Class 9

Punjab State Board PSEB 9th Class Maths Book Solutions Chapter 9 Areas of Parallelograms and Triangles Ex 9.3 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles Ex 9.3

Question 1.
In the given figure, E is any point on median AD of a ∆ ABC. Show that ar (ABE) = ar (ACE).

Answer:

As AD is a median of ∆ ABC, it divides ∆ ABC into two triangles with equal areas.
∴ ar (ADB) = ar (ADC) …………… (1)
In ∆ ABC, AD is a median.
∴ BD = CD
Draw a line l through E and parallel to BC. Then, ∆ EDB and ∆ EDC are on the equal bases and between the same parallels l and BC.
∴ ar (EDB) = ar (EDC) ……………… (2)
Subtracting (2) from (1),
ar (ADB) – ar (EDB) = ar (ADC) – ar (EDC)
∴ ar (ABE) = ar (ACE)

Question 2.
In a triangle ABC, E is the midpoint of median AD. Show that ar (BED) = \(\frac{1}{4}\)ar (ABC).
Answer:

In ∆ ABC, AD is a median
∴ ar (ADB) = ar (ADC) = \(\frac{1}{2}\)ar (ABC)
Now, in ∆ ABD, E is the midpoint of AD.
∴ BE is a median of ∆ ABD.
∴ ar (BED) = ar (BEA) = \(\frac{1}{2}\)ar (ABD)
Thus, ar (BED) = \(\frac{1}{2}\)ar (ABD) and
ar (ABD) = \(\frac{1}{2}\)ar (ABC)
∴ ar (BED) = \(\frac{1}{2}\) × \(\frac{1}{2}\)ar (ABC)
∴ ar (BED) = \(\frac{1}{4}\)

Question 3.
Show that the diagonals of a parallelogram divide it into four triangles of equal area.
Answer:

The diagonals of a parallelogram bisect each other.
Hence, in parallelogram ABCD, diagonals AC and BD bisect each other at M. Thus, M is the midpoint of AC as well as BD.
In ∆ ABC, BM is a median.
∴ ar (ABM) = ar (CBM) ……………. (1)
In ∆ BCD, CM is a median.
∴ ar (CBM) = ar (CDM) ……………. (2)
In ∆ ABD, AM is a median.
∴ ar (ABM) = ar (DAM) ……………. (3)
From (1), (2) and (3),
ar (ABM) = ar (CBM) = ar (CDM) = ar (DAM)
Thus, the diagonals of a parallelogram divide it into four triangles of equal areas.

Question 4.
In the given figure, ABC and ABD are two triangles on the same base AB. If line segment CD is bisected by AB at O, show that ar (ABC) = ar (ABD).

Answer:
Drawing a line through A and parallel to CD, ∆ AOC and A AOD are on the equal bases and between the same parallels.
∴ ar (AOC) = ar (AOD) ……………… (1)
Similarly, drawing a line through B and parallel to CD, A BOC and A BOD are on the equal bases and between the same parallels.
∴ ar (BOC) = ar (BOD) ……………. (2)
Adding (1) and (2),
ar (AOC) + ar (BOC) = ar (AOD) + ar (BOD)
∴ ar (ABC) = ar (ABD)

Question 5.
D, E and F are respectively the midpoints of the sides BC, CA and AB of a A ABC. Show that:
(i) BDEF is a parallelogram.
(ii) ar (DEF) = \(\frac{1}{4}\)ar (ABC)
(iii) ar (BDEF) = \(\frac{1}{2}\)ar (ABC)
Answer:

In ∆ ABC, F and E are the midpoints of AB and AC respectively.
∴ FE || BC
∴ FE || BD
In ∆ ABC, E and D are the midpoints of AC and BC respectively.
∴ ED || AB
∴ ED || FB
In quadrilateral BDEF, FE || BD and ED || FB.
∴ Quadrilateral BDEF is a parallelogram. …… Result (i)
Similarly, quadrilaterals AFDE and FDCE are parallelograms.
In parallelogram BDEF, FD is a diagonal.
∴ ar (BDF) = ar (DEF) ……………….. (1)
In parallelogram AFDE, EF is a diagonal.
∴ ar (AFE) = ar (DEF) ……………….. (2)
In parallelogram FDCE, ED is a diagonal.
∴ ar (DCE) = ar (DEF) ………………. (3)
∆ ABC is made up of four non-overlapping triangles, ∆ BDF, ∆ AFE, ∆ DCE and ∆ DEF.
∴ ar (ABC)
= ar (BDF) + ar (AFE) + ar (DCE) + ar (DEF)
∴ ar (ABC)
= ar (DEF) + ar (DEF) + ar (DEF) + ar (DEF) [From (1), (2) and (3)]
∴ ar (ABC) = 4ar (DEF)
∴ ar (DEF) = \(\frac{1}{4}\)ar (ABC) …………. Result (ii)
Now, ar (BDEF) = ar (BDF) + ar (DEF)
∴ ar (BDEF) = ar (DEF) + ar (DEF)
∴ ar (BDEF) = 2ar (DEF)
∴ ar (BDEF) = 2 × \(\frac{1}{4}\)ar (ABC)
∴ ar (BDEF) = \(\frac{1}{2}\)ar (ABC) ……. Result (iii)

Question 6.
In the given figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that:
(i) ar (DOC) = ar (AOB)
(ii) ar (DCB) = ar (ACB)
(iii) DA || CB and ABCD is a parallelogram. [Hint: From D and B, draw perpendiculars to AC.]

Answer:
Draw DM ⊥ AC and BN ⊥ AC, where M and N are points on AC.
In ∆ DMO and ∆ BNO,
DO = BO (Given)
∠ DOM = ∠ BON (Vertically opposite angles)
∠ DMO = ∠ BNO (Right angles)
∴ ∆ DMO ≅ ∆ BNO (AAS rule)
∴ DM = BN (CPCT)
Now, in ∆ DMC and ∆ BNA,
DM = BN
hypotenuse DC = hypotenuse BA (Given)
∠ DMC = ∠ BNA (Right angles)
∴ ∆ DMC = ∆ BNA (RHS rule)
∴ ar (DMC) = ar (BNA) ……………… (1)
Moreover, ar (DMO) = ar (BNO) (∵ ∆ DMO ≅ ∆ BNO) ………………… (2)
Adding (1) and (2),
ar (DMC) + ar (DMO) = ar (BNA) + ar (BNO)
∴ ar (DOC) = ar (AOB) (Non-overlapping triangles) … Result (i)
Now, adding ar (COB) on both the sides,
ar (DOC) + ar (COB) = ar (AOB) + ar (COB)
∴ ar (DCB) = ar (ACB) ……… Result (ii)
∆ DCB and ∆ ACB are on the same base BC and their areas are equal.
∴ ∆ DCB and ∆ ACB are between the same parallels DA and CB.
∴ DA || CB ……………… (3)
∆ DMO = ∆ BNO gives OM = ON
∆ DMC = ∆ BNA gives CM = AN
∴ OM + CM = ON + AN
∴ OC = OA
Also, OD = OB (Given)
Thus, in quadrilateral ABCD, diagonals AC and BD bisect each other at O.
∴ ABCD is a parallelogram. ……………….. (4)
Taking (3) and (4) together,
DA || CB and ABCD is a parallelogram. ………. Result (iii)

Question 7.
D and E are points on sides AB and AC respectively of ∆ ABC such that ar (DBC) = ar (EBC). Prove that DE || BC.
Answer:

ar (DBC) = ar (EBC)
∴ ∆ DBC and ∆ EBC have equal areas.
Here, ∆ DBC and ∆ EBC are on the same base BC and their areas are equal.
∴ ∆ DBC and ∆ EBC are between the same parallels DE and BC.
∴ DE || BC

Question 8.
XY is a line parallel to side BC of a ‘triangle ABC. If BE II AC and CF II AB meet ( XY at E and F respectively, show that ar (ABE) = ar (ACF).
Answer:

Suppose line XY drawn parallel to BC intersects AB and AC at M and N respectively.
In quadrilateral EBCN, EN || BC and BE || CN.
∴ EBCN is a parallelogram.
In quadrilateral MBCF, MF || BC and BM || CF.
∴ MBCF is a parallelogram.
Parallelograms EBCN and MBCF are on the same base BC and between the same parallels BC and EF
∴ ar (EBCN) = ar (MBCF)
∴ ar (BME) + ar (MBCN) = ar (MBCN) + ar (CFN)
∴ ar (BME) = ar (CFN) ……………….. (1)
Now, ∆ BME and ∆ CFN are between the same parallels EF and BC and their areas are equal.
∴ Their bases are equal.
∴ EM = NF
Through A, draw line PQ parallel to EF.
Then, ∆ AEM and ∆ ANF are on equal bases s and between the same parallels PQ and EF. !>
∴ ar (AEM) = ar (ANF) ……………… (2)
Adding (1) and (2),
ar (BME) + ar (AEM) = ar (CFN) + ar (ANF)
∴ ar (ABE) = ar (ACF)

Question 9.
The side AB of a parallelogram ABCD is ‘produced to any point P. A line through A and parallel to CP meets CB produced at and then parallelogram PBQR is completed (see the given figure). Show that: ar (ABCD) = ar (PBQR)
[Hint: Join AC and PQ. Now compare ar (ACQ) and ar (APQ).]

Answer:
Join AC and PQ.
∆ CAQ and ∆ PAQ are on the same base AQ and between the same parallels CP and AQ.
∴ ar (CAQ) = ar(PAQ)
∴ ar (ACB) + ar (ABQ) = ar (PBQ) + ar (ABQ) (Non-overlapping triangles)
∴ ar (ACB) = ar (PBQ) ……………. (1)
In parallelogram ABCD, AC is a diagonal.
∴ ar (ACB) = \(\frac{1}{2}\)ar (ABCD) ……………… (2)
In parallelogram PBQR, PQ is a diagonal.
∴ ar (PBQ) = \(\frac{1}{2}\)ar (PBQR) ………………. (3)
From (1), (2) and (3),
\(\frac{1}{2}\)ar (ABCD) = \(\frac{1}{2}\)ar (PBQR)
∴ ar (ABCD) = ar (PBQR)

Question 10.
Diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar (AOD) = ar (BOC).

Answer:
∆ DAB and ∆ CBA are on the same base AB and between the same parallels AB and CD.
∴ ar (DAB) = ar (CBA)
∆ DAB and ∆ CBA are both formed by two non-overlapping triangles.
∴ ar (DAB) = ar (AOD) + ar (OAB) and
ar (CBA) = ar (BOC) + ar (OAB)
∴ ar (AOD) + ar (OAB) = ar (BOC) + ar (OAB)
∴ ar (AOD) = ar (BOC)

Question 11.
In the given figure, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Show that
(i) ar (ACB) = ar (ACF)
(ii) ar (AEDF) = ar (ABCDE)

Answer:
∆ ACB and ∆ ACF are on the same base AC and between the same parallels AC and BF.
∴ ar (ACB) = ar (ACF)
Adding ar (AEDC) on both the sides,
ar (ACF) + ar (AEDC) = ar (ACB) + ar (AEDC)
∴ ACF and quadrilateral AEDC are two non-overlapping figures and they form quadrilateral AEDF.
Similarly, ∆ ACB and quadrilateral AEDC are two non-overlapping figures and they form pentagon ABCDE.
∴ ar (AEDF) = ar (ABCDE)

Question 12.
A villager Itwaari has a plot of land of the shape of a quadrilateral. The Gram Panchayat of the village decided to take over some portion of his plot from one of the comers to construct a Health Centre. Itwaari agrees to the above proposal with the condition that he should be given equal amount of land in lieu of his land adjoining his plot so as to form a triangular plot. Explain how this proposal will be implemented.
Answer:

Suppose, quadrilateral ABCD is the plot of Itwaari in the shape of a quadrilateral.
Draw diagonal AC. Through D draw a line parallel to AC to intersect BC produced at P
Join PA to intersect CD at Q.
Here, ∆ DAC and ∆ PAC are on the same base AC and between the same parallels DP and AC.
∴ ar (DAC) = ar (PAC)
∴ ar (DAQ) + ar (QAC) = ar (PQC) + ar (QAC)
∴ ar (DAQ) = ar (PQC)
Thus, we get two triangles – ∆ DAQ and ∆ PQC of equal areas.
So, the Gram Panchayat should take triangular region DAQ from the plot of Itwaari and in exchange give him triangular region PQC. Then, the plot of Itwaari will be of the triangular shape PAB and its area will be the same as the area of original plot ABCD.

Question 13.
ABCD is a trapezium with AB || DC. A line parallel to AC intersects AB at X and BC at Y. Prove that ar (ADX) = ar (ACY).
[Hint: Join CX.)
Answer:

Join CX.
∆ ADX and ∆ ACX are on the same base AX and between the same parallels AB and CD.
∴ ar (ADX) = ar (ACX) …………… (1)
A ACX and A ACY are on the same base AC and between the same parallels AC and XY.
∴ ar (ACX) = ar (ACY) ……………. (2)
From (1) and (2),
ar (ADX) = ar (ACY)

Question 14.
In the given figure, AP || BQ || OR. Prove that ar (AQC) = ar (PBR).

Answer:
∆ CBQ and ∆ RBQ are on the same base BQ and between the same parallels BQ and CR.
∴ ar (CBQ) = ar (RBQ) ……. (1)
∆ ABQ and ∆ PBQ are on the same base BQ and between the same parallels BQ and AE
∴ ar (ABQ) = ar (PBQ) ……… (2)
Adding (1) and (2),
ar (CBQ) + ar (ABQ) = ar (RBQ) + ar (PBQ)
∴ ar (AQC) = ar(PBR) (Non-overlapping triangles)

Question 15.
Diagonals AC and BD of a quadrilateral ABCD intersect at O in such a way that ar (AOD) = ar (BOC). Prove that ABCD is a , trapezium.
Answer:

ar (AOD) = ar (BOC)
Adding ar (OAB) on both the sides,
ar (AOD) + ar (OAB) = ar (BOC) + ar (OAB)
∴ ar (DAB) = ar (CAB)
Thus, ∆ DAB and ∆ CAB are on the same base AB and their areas are equal.
Hence, by theorem 9.3, ∆ DAB and ∆ CAB are between the same parallel.
∴ DC || AB
In quadrilateral ABCD, DC || AB.
Hence, ABCD is a trapezium.

Question 16.
In the given figure, ar (DRC) = ar (DPC) and ar (BDP) = ar (ARC). Show that both the quadrilaterals ABCD and DCPR are trapeziums.

Answer:
ar (DRC) = ar (DPC)
Thus, ∆ DRC and ∆ DPC are on the same base DC and their areas are equal.
Hence, by theorem 9.3, they are between the same parallels.
∴ DC || RP
In quadrilateral DCPR, DC || RE .
∴ Quadrilateral DCPR is a trapezium.
Now, ar (BDP) = ar (ARC)
∴ ar (DPC) + ar (BDC) = ar (DRC) + ar (ADC) (Non-overlapping triangles)
But, ar (DPC) = ar (DRC)
∴ ar (BDC) = ar (ADC)
Thus, ∆ BDC and ∆ ADC are on the same base DC and their areas are equal.
Hence, by theorem 9.3, they are between the same parallels.
∴ AB || DC
In quadrilateral ABCD, AB || DC.
∴ Quadrilateral ABCD is a trapezium.

Punjab State Board PSEB 9th Class English Book Solutions English Vocabulary Antonyms Exercise Questions and Answers, Notes.

PSEB 9th Class English Vocabulary Antonyms

Match the Antonyms correctly :

A — B
admit — destroy
public — straight
junior — conclude
bent — stale
defend — senior
dark — private
fresh — well-lit
create — descend
ascend — dangerous
safe — attack
start — modern
ancient — deny
Answer:
admit → deny; public → private ; junior → senior; bent → straight; defend → attack; dark → well-lit; fresh → stale ; create → destroy; ascend → descend; safe → dangerous; start → conclude ; ancient → modem.

Complete the following sentences with the Antonyms given in the box. The italicized words will help you to choose the right words:

modern;  spend; withdraw; vacant; remember; success; punctual; expensive; plus; import

1. Ramesh, you are always ………… but why are you late today?
2. We will not ………… all the money now. We will save it for the rainy day.
3. Seven seats are still ………… They were all full by now last year.
4. I will deposit the whole amount. I can ………… it at any time I want to.
5. The Goyals ………… wool from Australia, make garments and export them to the U.K.
6. Don’t forget your sweater here ………….. to wear it when it is cold in Srinagar.
7. ……….. and failure go hand in hand in life.
8. This umbrella is very ……… I need a cheap one.
9. Rome has both ancient and ……….. buildings.
10. Can you tell me the minus and ……….. points of this proposal?
Answer:
1. punctual
2. spend
3. vacant
4. withdraw
5. import
6. Remember
7. Success
8. expensive
9. modem
10. plus.

Antonyms

जिन दो शब्दों के अर्थ एक-दूसरे के विपरीत हों, उन्हें Antonyms कहते हैं; जैसे
clean → dirty
deep → shallow
sorrow → joy

List Of Antonyms

Punjab State Board PSEB 9th Class English Book Solutions English Vocabulary Synonyms Exercise Questions and Answers, Notes.

PSEB 9th Class English Vocabulary Synonyms

Match the words in column A with their synonyms in column B :

A — B
just — reach
leave — wrath
permit — refuge
brave — fearless
certain — vanish
disappear — sure
anger — prevent
forbid — fair
shelter — inactive
attempt — depart
lazy — allow
arrive — try
Answer:
just = fair ; leave = depart ; permit = allow ; brave = fearless ; certain = sure ; disappear = vanish; anger = wrath ; forbid = prevent; shelter = refuge; attempt = try; lazy = inactive; arrive = reach.

Complete the sentences with the words given in the box. The italicized words will help you to choose the right words 

inactive ; pouring ; bright ; finish ; annoyed ; enormous ; collect ; wear ; try

1. Is the new student intelligent ? Yes, he is very ……………..
2. It was raining heavily. In fact, it was ………………
3. The show stopped at 10 p.m. because the policemen asked us to ………….. early.
4. She was very angry with my brother. I don’t know why she was so …………….. with him.
5. It was a very huge animal. We had never seen such an ……………. animal before.
6. He likes to assemble photographs. He has decided to …………….. about one thousand rare ones.
7. Don’t put on the red cap, …………….. the black one.
8. She is quite lazy. I don’t know why she is so ……………..
Answer:
1. bright
2. pouring
3. finish
4. annoyed
5. enormous
6. collect
7. wear
8. inactive.

Synonyms

जिन शब्दों का अर्थ एक ही हो अथवा मिलता-जुलता हो, उन्हें Synonyms कहते हैं; जैसे
hard = difficult
pray = request
fact = truth

List Of Synonyms

Punjab State Board PSEB 9th Class English Book Solutions English Vocabulary Words as different Parts of Speech Exercise Questions and Answers, Notes.

PSEB 9th Class English Vocabulary Words as different Parts of Speech

Write n’ for noun, ‘adj.’ for adjective, ‘adv.’ for adverb and ‘ for verb for the italicized word given in the sentences :

1. One must travel light while travelling by air.
2. This clock is ten minutes fast.
3. The chief guest’s speech was brief and to the point.
4. The officer was briefed about the facts of the case.
5. This bicycle cost him twelve hundred rupees.
6. Manvinder ran fast enough to reach the school in time.
7. What is your examiner’s schedule for tomorrow ?
8. You can buy many fancy items from Meena Bazar.
9. That tastes real good. Where did you get it from ?
10. What is the cost of this pair of socks ?
Answer:
1. adv.
2. adj.
3. adj.
4. v
5. v
6. adv.
7. n
8. adj.
9. adv.
10. n.

Use the given words in sentences as directed :

1. Fare (noun) – He has not paid the bus fare.
Fare (verb) – Ram has fared well in the examination.

2. Bear (noun) – There are many types of bear in the zoo.
Bear (verb) – He has borne much suffering in his life.

3. Wound (noun) – My wound has healed.
Wound (verb) – He was wounded seriously.

4. Round (adjective) – The earth is round.
Round (preposition)- The earth goes round the sun.
Round (adverb) – Everybody joins hands and dances round.

5. Fast (adjective) – He is a fast friend of mine.
Fast (adverb) – She is running very fast.
Fast (noun) – I observe fast on every Monday.

6. Stand (noun) – I saw him near the cycle-stand.
Stand (verb) – Will you please stand up?

7. Produce (noun) – This shop sells only fresh local produce.
Produce (verb) – She produced a hot meal for us within 20 minutes.

एक ही शब्द को भाषा के विभिन्न रूपों में इस्तेमाल किया जा सकता है। उदाहरण के रूप में निम्नलिखित वाक्यों को देखिए

They fought to the last. (Noun)
The war lasted one month. (Verb)
He was the last man to come. (Adjective)
He spoke last at the meeting. (Adverb)

यहां शब्द ‘last’ को पहले वाक्य में Noun के रूप में, दूसरे वाक्य में एक Verb के रूप में, तीसरे वाक्य में एक Adjective तथा चौथे वाक्य में एक Adverb के रूप में इस्तेमाल किया गया है।
Examples :

Act
Verb : He acted upon my advice.
Noun : It was an act of kindness.

All
Noun : I lost my all in this business.
Adjective : All the boys reached the playground.

Bail
Verb : He will be bailed out today.
Noun : He was set free on bail.

Back
Verb : He becked my proposal.
Noun: He has pain in his back.
Adjective : He came through the back door.

Better
Adjective : This book is better than that.
Adverb: He fared better in the test than he had hoped.
Verb : We hope to better the conditions of our workers.
Noun (pl.) : Follow your betters.

Book
Verb : Get your luggage booked.
Noun : I have read this book.

Close
Noun : The meeting came to a close in time.
Verb : The school is closed for summer vacation.

Dawn
Verb : The truth dawned upon him yesterday.
Noun : I always get up before dawn.

Drive
Verb : Suresh is driving the car at top speed.
Noun : He had a good drive in the morning.

Effect
Verb : The prisoner effected his escape.
Noun : My advice had the desired effect.

Fare
Verb : I have fared badly in my test.
Noun : We paid the bus fare.

Fix
Noun : We are in a fix now.
Verb : Let us fix the programme.

Face :
Verb: He has faced many difficulties in his life.
Noun Wash your face.

Fast
Noun: They keep a fast every Friday.
Adjective: This is a fast train.
Verb: I fast once every week.

Hand
Verb: The thief was handed over to the police.
Noun: Wash your hands.

Iron
Verb: Iron your clothes.
Noun: Iron is a useful metal.
Adjective: This knife has an iron handle.

Idle :
Verb: Do not idle away your precious time.
Noun: The idle fail in their life.
Adjective: Sohan is an idle boy.

Like
Verb: I like her ways.
Noun: Everybody has his likes and dislikes.
Adjective: Like poles repel each other.

Light
Noun: There was no light in the room.
Adjective: Always take a light diet to keep fit.
Verb: Light the lamp.
Adverb: Travel light if you must.

Less
Adjective: He is paying less attention to studies these days.
Verb: He is less intelligent than his brother.
Noun: He won’t be satisfied with less.

Near :
Adjective: He is a near relation of the headmaster.
Verb: He is nearing his end.
Adverb: Come near

Right
Noun: We must fight for our rights.
Adjective: This is my right hand.
Verb: Every wrong should be righted.

Round
Noun: Let us have a round of cards.
Adjective: The earth is round.
Verb: He rounded his lips in anger.
Adjective: He turned round.
Preposition: She wore a necklace round her neck.

Second :
Verb: I will second your proposal.
Noun: I will come back in a second.
Adjective: February is the second month of the year.

Still
Noun: In the still of the night, a thief entered our house.
Adjective: The night was still.

Stone
Adjective: It is a stone wall.
Verb: The boys stoned the dog to death.
Noun: The boys threw stones at the beggar.

Time
Noun: What is the time by your watch ?
Verb: The train is timed to reach here at 5 p.m.

Well
Noun: We have a well in our garden.
Adjective: The patient is now well.
Adverb: He has fared well in the examination.
Verb: Tears welled up in her eyes.

While
Noun: Rest a little while.
Verb: He is whiling away his time.

Will
Noun: Where there is a will, there is a way.
Verb: Have what you will.

Punjab State Board PSEB 9th Class Maths Book Solutions Chapter 9 Areas of Parallelograms and Triangles Ex 9.2 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles Ex 9.2

Question 1.
In the given figure, ABCD is a parallelogram, AE ⊥ DC and CF ⊥ AD. If AB = 16 cm, AE = 8 cm and CF = 10 cm, find AD.

Answer:
The area of a parallelogram is the product of its base and the altitude corresponding to that base.
Here, in parallelogram ABCD, the altitude corresponding to base DC is AE and the altitude corresponding to base AD is CF.
∴ ar(ABCD) = DC × AE = AD × CF
∴ DC × AE = AD × CF
∴ AB × AE = AD × CF (∵ AB = DC In parallelogram ABCD)
∴ 16 × 8 = AD × 1O
∴ AD = \(\frac{16 \times 8}{10}\)
∴ AD = \(\frac{128}{10}\)
∴ AD = 12.8 cm

Question 2.
If E, F, G and H are respectively the midpoints of the sides of a parallelogram ABCD, show that ar(EFGH) = \(\frac{1}{2}\)ar (ABCD).
Answer:
In parallelogram ABCD. E, F, G and H are the midpoints of AB, BC. CD and DA respectively.
Draw GE.

In parallelogram ABCD, AB || CD and AB = CD.
∴ BE || CG and BE (\(\frac{1}{2}\) AB) = CG (\(\frac{1}{2}\) CD)
∴ Quadrilateral EBCG is a parallelogram.
∴ GE || BC
Now, ∆ EFG and parallelogram EBCG are on the same base GE and between the same parallels GE and BC.
∴ ar(EFG)= \(\frac{1}{2}\)ar (EBCG) ………………. (1)
Similarly, ∆ EHG and parallelogram AEGD are on the same base GE and between the same parallels GE and DA.
∴ ar(EHG) = \(\frac{1}{2}\)ar (AEGD) ……………….. (2)
Adding (1) and (2),
ar (EFG) + ar (EHG) = \(\frac{1}{2}\) ar (EBCG) + \(\frac{1}{2}\) ar (AEGD)
∴ ar (EFGH) = \(\frac{1}{2}\) [ar (EBCG) + ar (AEGD)]
∴ ar (EFGH) = \(\frac{1}{2}\) ar (ABCD)

Question 3.
p and Q are any two points lying on the sides DC and AD respectively of a parallelogram ABCD. Show that ar (APB) = ar (BQC).

Answer:
Here, ∆ APB and parallelogram ABCD are on the same base AB and between the same parallels AB and CD.
∴ ar (APB) = \(\frac{1}{2}\)ar (ABCD) …………… (1)
Similarly, ∆ BQC and parallelogram ABCD are on the same base BC and between the same parallels BC and DA.
∴ ar (BQC) = \(\frac{1}{2}\)ar (ABCD) ……………… (2)
From (1) and (2),
ar(APB) = ar(BQC)

Question 4.
In the given figure, P is a point in the interior of a parallelogram ABCD. Show that,
(i) ar (APB) + ar (PCD) = \(\frac{1}{2}\)ar (ABCD)
(ii) ar (APD) + ar (PBC) = ar (APB) + ar (PCD)
[Hint: Through P, draw a line parallel to AB.]

Answer:
Through P, draw a line parallel to AB which intersects BC at Q and AD at R.
Now, in quadrilateral ABQR,
AB || QR (By construction)
BQ || AR (In parallelogram ABCD, BC || AD)
∴ Quadrilateral ABQR is a parallelogram.
Similarly, DCQR is a parallelogram.
∆ APB and parallelogram ABQR are on the same base AB and between the same parallels AB and QR.
∴ ar(APB) = \(\frac{1}{2}\)ar (ABQR) ……………….. (1)
Similarly, ∆ PCD and parallelogram DCQR are on the same base DC and between the same parallels DC and QR.
∴ ar(PCD) = \(\frac{1}{2}\)ar (DCQR) ………………… (2)
Adding (1) and (2),
= ar(APB) + ar (PCD)
= \(\frac{1}{2}\)ar (ABQR) + \(\frac{1}{2}\)ar (DCQR)
∴ ar (APB) + ar (PCD)
= \(\frac{1}{2}\) [ar (ABQR) + ar (DCQR)]
∴ ar (APB) + ar (PCD) = \(\frac{1}{2}\)ar (ABCD) …………………. (3)
Now, through P, draw a line parallel to AD which intersects AB at S and CD at T.
Then, as above, it can be proved that
ar(APD) + ar(PBC) = \(\frac{1}{2}\)ar (ABCD) ……………………. (4)
From (3) and (4),
ar (APD) + ar (PBC) = ar (APB) + ar (PCD)

Question 5.
In the given figure, PQRS and ABRS are parallelograms and X is any point on side BR. Show that:
(i) ar (PQRS) = ar (ABRS)
(ii) ar(AXS) = \(\frac{1}{2}\)ar (PQRS)

Answer:
Parallelograms PQRS and ABRS are on the same base RS and between the same parallels PB and SR.
∴ ar (PQRS) = ar (ABRS) …………….. (1)
∆ AXS and parallelogram ABRS are on the same base AS and between the same parallels AS and BR.
∴ ar (AXS) = \(\frac{1}{2}\)ar (ABRS) ………………… (2)
From (1) and (2),
ar (AXS) = \(\frac{1}{2}\)ar (PQRS)

Question 6.
A farmer was having a field in the form of a parallelogram PQRS. She took any point A on RS and joined it to points P and Q. In how many parts the field Is divided? What are the shapes of these parts? The farmer wants to sow wheat and pulses in equal portions of the field separately. How should she do it?
Answer:

By taking any point A on RS and joining it to points P and Q, the field is divided into three triangular parts as ∆ PSA, ∆ APQ and ∆ QRA.
Here, ∆ APQ and parallelogram PQRS are on the same base PQ and between the same parallels PQ and SR.
∴ ar (APQ) = \(\frac{1}{2}\)ar (PQRS)
∴ ar (PSA) + ar(QRA) = \(\frac{1}{2}\)ar(PQRS)
Thus, ar (APQ) = ar (PSA) + ar (QRA)
Now, as the farmer wants to sow wheat and pulses in equal portions of the field separately.
she has two options as below to do so:
(1) She should sow wheat in ∆ APQ and pulses in ∆ PSA as well as ∆ QRA.
(2) She should sow pulses in ∆ APQ and wheat in ∆ PSA as well as ∆ QRA.

Punjab State Board PSEB 9th Class English Book Solutions English Vocabulary Nature of Words Exercise Questions and Answers, Notes.

PSEB 9th Class English Vocabulary Nature of Words

Read the words and phrases that convey different meanings of the word ‘set given below :

Fixed, started, a group of same things, arranged, written or done, a piece of equipment, put. Now write the word / group of words that give the meaning of ‘set’ against each sentence.

1. My mother gifted a set of handkerchiefs to my younger sister.
2. The jewels were set beautifully in gold.
3. Raghu is well set in his new job.
4. Several TV sets have been stolen from his shop.
5. Let’s start wearing woollen clothes; the winter has set in.
6. The date of the test has not been set yet.
7. She has set the alarm for 5 o’clock, so that she is not late.
8. The surgeon who set my uncle’s hip bone is very popular.
9. Old people have set views about everything.
10. The table has been set for the tea party.
Answer:
1. a group of same things
2. fixed
3. settled
4. a piece of equipment
5. started
6. fixed
7. put fixed
9. fixed
10. arranged.

किसी एक शब्द के विभिन्न प्रयोगों के विभिन्न अर्थ हो सकते हैं। उदाहरण के रूप में नीचे दिए वाक्यों में run शब्द के आथों को देखिए

1. He can run very fast and win the prize. (race)
2. Don’t you think that the film will run at least for 6 weeks? (continue)
3. Suman runs from one table to another to finish her work in time. (hurries)
4. It is strange that Mr. Sahay is able to run two businesses effortlessly. (manages)
5. Our school will run Spoken English classes during the summer vacation. (provide)
6. Trains often run late in winter. (travel)
7. I can’t run on my small salary. (afford)
8. When she peels onions, tears run down her cheeks. (flow)

Punjab State Board PSEB 9th Class English Book Solutions English Grammar Tenses (Concord and Sequence) Exercise Questions and Answers, Notes.

PSEB 9th Class English Grammar Tenses (Concord and Sequence)

Correct the tense in the following sentences :

(a)
1. They have come yesterday.
2. I have bought this house last year.
3. I am the person who have saved your life.
4. I shall call you when the lunch will be ready.
5. Dinesh will come if you will invite him.
6. I had been to Delhi recently.
7. She is one of those persons who never tells a lie.
8. She is working in the office for two years.
9. I had gone to the club last night.
10. I have written to him yesterday.
Answer:
1. They came yesterday.
2. I bought this house last year.
3. I am the person who has saved your life.
4. I shall call you when the lunch is ready.
5. Dinesh will come if you invite him.
6. I have been to Delhi recently.
7. She is one of those persons who never tell a lie.
8. She has been working in the office for two years.
9. I went to the club last night.
10. I wrote to him yesterday.

(b) 1. I hoped that my friend will help me
2. Take care that you will not be cheated.
3. I know them for the last four years.
4. The patient died before the doctor arrived.
5. Newton has discovered the Law of Gravitation.
6. I am studying hard for the last three weeks.
7. He is suffering from fever since Monday.
8. I had visited Shimla last summer.
9. I am sorry I did not write that letter yet.
10. His younger sister is died.
Answer:
1. I hoped that my friend would help me.
2. Take care that you are not cheated.
3. I have been knowing them for the last four years.
4. The patent had died before the doctor arrived.
5. Newton discovered the Law of Gravitation.
6. I have been studying hard for the last three weeks.
7. He has been suffering from fever since Monday.
8. I visited Shimla last summer.
9. I am sorry I have not written that letter yet.
10. His younger sister is dead.

Put the correct Verbs in the blanks :

1. The tallest of these boys ……….. next door to me. (live, lives)
2. Slow and steady ……….. the race. (win, wins)
3. Neither Ashok nor Prem ………. any right to the property. (has, have)
4. Neither the captain nor the soldiers ……….. been arrested. (has, have)
5. Neither he nor I ……….. money to spare for this. (has, have)
6. You, as well as he, ……….. innocent. (is, are)
7. None but the brave ……….. the fair. (deserve, deserves)
8. The United States ……… a prosperous country. (is, are)
9. The committee ………. divided in their opinion. (was, were)
10. I, who ……….. your friend, will stand by you. (is, am, are)
Answer:
1. lives
2. wins
3. has
4. have
5. have
6. are
7. deserves
8. is
9. were
10. am.

Put the correct Verbs in the blanks :

1. All the players in my team ………. done well. (has, have)
2. Bread and butter ……… what they want. (is, are)
3. Either Sushil or Rakesh ……….. done this mischief…….. (has, have)
4. Neither he nor his servants ……….. honest. (was, were)
5. Neither you nor Sham ……….. to be capable of doing this. (appear, appears)
6. Not only the workman but the supervisor also ………. been dismissed. (has, have)
7. Each day and each hour ……….. its own importance. (has, have)
8. The news of the flood ……….. caused great anxiety. (has, have)
9. The jury ……….. divided in their opinion. (was, were)
10. This is one of the best novels that ………… been published this year. (has, have)
Answer:
1. have
2. is
3. has
4. were
5. appears
6. has
7. has
8. has
9. were
10. have.

Put the correct Verbs in the blanks :

1. The cost of all types of pens ………. gone up. (has, have)
2. Time and tide ……….. for none ……….. (wait, waits)
3. Either you or he ……….. mistaken. (is, are)
4. Either Rajinder or his parents ……… responsible for this. (was, were)
5. Neither you nor I ……… lucky. (am, are)
6. He, as well as you, ……….. to blame. (is, are)
7. None of the ships ……….. rescued from the storm. (was, were)
8. The Arabian Nights ……….. interesting stories. (contain, contains)
9. The crew did ……….. best to save the ship. (its, their)
10. This is one of the most difficult lessons that ……… been taught. (has, have)
Answer:
1. has
2. wait
3. is
4. were
5. am
6. is
7. was
8. contains
9. their
10. have.

Put the correct Verbs in the blanks :

1. The toys that were bought by my son……….. really useful. (is, are)
2. Oil and water ……….. mix. (does not, do not)
3. Neither the judge nor the witnesses ……… him. (believe, believes)
4. Neither the Principal nor the lecturers ……….. present at the meeting. (was, were)
5. Either he or I ……….. wrong. (am, are)
6. The workmen, with their leaders ………. been arrested. (have, has)
7. One of my friends ………. the owner of this factory. (is, are)
8. Good crockery ……….. expensive. (is, are)
9. The jury ……….. divided on the issue. (was, were)
10. I am the one who ………. always stood for justice. (has, have)
Answer:
1. are
2. do not
3. believe
4. were
5. am
6. have
7. is
8. is
9. were
10. has.

Put the correct Verb in the blanks :

1. He ……….. regularly. (work, works)
2. Tobacco and alcohol ………. injurious to health. (is, are)
3. Either the officer or the assistants ………. responsible for this error. (is, are)
4. Me or his friends ………. to blame. (was, were)
5. Neither my brother nor I ……….. ever been to Mumbai. (has, have)
6. The gallery, with its beautiful pictures, ………. a great attraction. (is, are)
7. Either of these two proposals ……….. acceptable to me. (is, are)
8. The West Indies ……….. a land of great cricketers. (is, are)
9. The Assembly ……….. in session.
10. He is one of those leaders who …………. always ready to do or die.
Answer:
1. works
2. are
3. are
4. were
5. have
6. is
7. is
8. are
9. is
10. are.

Use Of Tenses

Tense (काल)-Verb के जिस रूप से यह पता चले कि क्रिया कब हुई है, उसे Tense कहा जाता है। Tense तीन प्रकार के होते हैं

1. Present Tense (वर्तमान काल)
2. Past Tense (भूत काल)
3. Future Tense (भविष्य काल)
प्रत्येक Tense के चार भिन्न-भिन्न रूप होते हैं

1. (i) Present Indefinite Tense (or) Simple Present
(ii) Present Continuous Tense
(iii) Present Perfect Tense
(iv) Present Perfect Continuous Tense.

2. (i) Past Indefinite Tense (or) Simple Past
(ii) Past Continuous Tense
(iii) Past Perfect Tense
(iv) Past Perfect Continuous Tense.

3. (i) Future Indefinite (or) Simple Future
(ii) Future Continuous Tense
(iii) Future Perfect Tense
(iv) Future Perfect Continuous Tense.

(A) Present Tense

1. Present Indefinite Tense प्रयोग – इस Tense का प्रयोग निम्नलिखित स्थितियों में किया जाता है
1. सदा सत्य बातों के लिए:
(The sun shines by day, and the moon by night.)

2. चरित्र-सम्बन्धी स्थायी आदतों के लिए –
(He never tells a lie.)

3. वास्तविक Present और निकट भविष्य के लिए –
1. I understand what you say.
2. He leaves for Mumbai tomorrow.

नियम-

• इस Tense में Verb की पहली फार्म का प्रयोग किया जाता है।
• यदि subject एकवचन हो, तो verb की पहली फार्म के साथ या का प्रयोग किया जाता है।
• Negative तथा Interrogative वाक्यों में Helping Verb के रूप में do / does का प्रयोग किया जाता है।
• बहुवचन subject के लिए do का, एकवचन subject के लिए does का प्रयोग किया जाता है। Does के साथ प्रयोग की जाने वाली Verb की पहली फार्म के साथ s या es नहीं लगाया जाता है।
• I तथा you के लिए प्रयोग की जाने वाली Verb की पहली फार्म के साथ s या es नहीं लगाया जाता है। इनके लिए बहुवचन subject की भान्ति केवल Verb की पहली फार्म का ही प्रयोग किया जाता है।

2. Present Continuous Tense प्रयोग-इस Tense का प्रयोग निम्नलिखित स्थितियों में किया जाता है

1. बोलने या लिखने के समय चल रही किसी घटना का वर्णन करने के लिए; जैसे (He is writing a letter:)
2. किसी आदत अथवा व्यवहार का वर्णन करने के लिए भी कभी-कभी इस Tense का प्रयोग किया जाता है; जैसे
(She is working regularly these days.) नियम- इस Tense के वाक्यों में is / am / are लगा कर Verb के ing. वाले रूप का वर्णन किया जाता है।
Subject + is / am / are + Verb का ing वाला रूप + ………….

Note : Present Continuous Tense तथा Present Indefinite Tense के अन्तर को अच्छी प्रकार समझ लेना चाहिए। निम्नलिखित नियम याद रखिए
Present Indefinite Tense — for habitual actions; for universal truths; for actions going to take place in near future.
Present Continuous Tense — for actions in progress at the time of speaking.

3. Present Perfect Tense प्रयोग-इस Tense का प्रयोग अभी-अभी (वर्तमान में) पूरी हुई किसी घटना का वर्णन करने के लिए
किया जाता है। नियम- इस Tense के वाक्यों में has / have लगा कर Verb की तीसरी फार्म का प्रयोग किया जाता है; जैसे
1. She has just finished her work.
2. I have spent all my money on books.

4. Present Perfect Continuous Tense प्रयोग- इस Tense का प्रयोग तब किया जाता है जब कोई क्रिया भूत काल में आरम्भ हुई हो और अब भी चल रही हो; जैसे
1. He has been reading English for two years.
2. They have been reading this book since morning.

नियम-
(1) इस Tense के वाक्यों में has / have + been + V₁-ing + since / for का प्रयोग किया जाता है।
(2) For का प्रयोग तब किया जाता है जब क्रिया के जारी रहने की अवधि (period of time) दी हुई हो; जैसे for fifteen minutes; for half an hour; for a month; for three years ; for a long time.
(3) Since का प्रयोग तब किया जाता है जब क्रिया के आरम्भ होने का निश्चित समय (point of time) दिया हुआ हो; जैसे
since Sunday; since breakfast; since yesterday ; since Diwali; since January 1998; since the first of May.

(B) Past Tense

1. Past Indefinite Tense प्रयोग- इस Tense का प्रयोग तब किया जाता है जब कोई घटना भूत काल में आरम्भ होकर भूत काल में ही समाप्त हो गई हो। (We learnt English at school.)

नियम-
(1) इस Tense के affirmative वाक्यों में Verb की दूसरी फार्म का प्रयोग किया जाता है।
(2) Negative वाक्यों में did not + Verb की पहली फार्म का प्रयोग किया जाता है।
(3) Interrogative वाक्यों में Did + subject + Verb की पहली फार्म का प्रयोग किया जाता है।
(4) Did का प्रयोग करने के बाद सदा Verb की पहली फार्म का ही प्रयोग किया जाता है।

2. Past Continuous Tense प्रयोग – इस Tense का प्रयोग तब किया जाता है जब कोई क्रिया भूतकाल में किसी निश्चित समय पर चल रही हो। (He was writing a letter when I saw him.)
नियम – इस Tense के वाक्यों में was / were लगा कर Verb के ing वाले रूप का प्रयोग किया जाता है।

Note : Past Continuous Tense तथा Past Indefinite Tense के अन्तर को अच्छी प्रकार से समझ लेना चाहिए। निम्नलिखित नियम याद रखिए
Past Indefinite Tense – for completed activity.
Past Continuous Tense – for continuity in past.

3. Past Perfect Tense प्रयोग – इस Tense का प्रयोग तब किया जाता है यदि कोई क्रिया भूतकाल में किसी निश्चित समय (moment) से पहले या किसी निश्चित समय तक पूरी हो चुकी हो।
1. The patient had died before the doctor came.
2. I had finished my work by evening.

नियम – इस Tense में had लगा कर Verb की तीसरी फार्म का प्रयोग किया जाता है।

Note : यदि किसी वाक्य में भूत काल में घटी हुई दो घटनाओं का वर्णन हो, तो उनमें से जो घटना पहले घटी हो उसके लिए Past Perfect Tense का प्रयोग किया जाता है, और जो घटना अपेक्षाकृत बाद में घटी हो उसके लिए Past Indefinite Tense का प्रयोग किया जाता है।

4. Past Perfect Continuous Tense प्रयोग – इस Tense का प्रयोग तब किया जाता है यदि कोई क्रिया भूतकाल में आरम्भ होकर भूतकाल में ही किसी निश्चित समय पर चल रही हो; जैसे(When I reached there at 2 p.m., he had been waiting for me since 1:30 p.m.) नियम – इस Tense के वाक्यों में had + been + V₁-ing + since / for का प्रयोग किया जाता

(C) Future Tense

Future Indefinite Tense

प्रयोग- इस Tense का प्रयोग साधारण रूप में भविष्य में होने वाली किसी घटना का वर्णन करने के लिए किया जाता है; जैसे–
(The post will arrive late. We shall start early.)
नियम – इस Tense के वाक्यों में will / shall लगा कर Verb की पहली फार्म का प्रयोग किया जाता है।
Note : Will और Shall सम्बन्धी प्रयोग के लिए निम्नलिखित नियम ध्यान में रखिए

(1) Assertive वाक्यों में
1. यदि किसी भविष्य की घटना का केवल साधारण रूप से ही वर्णन हो, तो-
First Person के लिए shall का प्रयोग किया जाता है।
Second और Third Persons के लिए will का प्रयोग किया जाता है।

2. यदि वाक्य में किसी भविष्य-सम्बन्धी आदेश (command), प्रण (promise), निश्चय (determination), धमकी (threat) आदि का वर्णन हो, तो First Person के लिए will का प्रयोग किया जाता है।

3. यदि वक्ता अपने किसी भविष्य-सम्बन्धी उद्देश्य (intention) को प्रकट करना
चाहता हो तो वह अपने लिए Will का प्रयोग करेगा। अर्थात् ऐसे वाक्यों में First person के साथ will का प्रयोग किया जाता है।

(2) Interrogative वाक्यों में
1. प्रश्न-वाचक वाक्यों में प्राय: Will I और Shall you का प्रयोग नहीं किया जाता है।
2. Third Person के लिए will अथवा shall किसी का भी प्रयोग किया जा सकता है।
Shall he – भविष्य सम्बन्धी आदेश आदि के लिए।
Will he – भविष्य सम्बन्धी साधारण क्रिया के लिए।

Future Continuous Tense –

प्रयोग-इस Tense का प्रयोग तब किया जाता है जब भविष्य में किसी विशेष समय का वर्णन हो, और कोई क्रिया उस विशेष समय से पहले आरम्भ होकर उसके बाद में पूरी होनी हो; जैसे
(We shall be waiting for you when you get back tomorrow).
नियम – इस Tense के वाक्यों में will / shall के साथ be लगा कर Verb के ing वाले रूप का प्रयोग किया जाता है।

Note :
Future Continuous Tense तथा Future Indefinite Tense के अन्तर को अच्छी तरह से समझ लेना चाहिए। निम्नलिखित नियम याद रखिए
Future Indefinite Tense – for an action which begins at a certain future moment.
Future Continuous Tense – for an action which begins before and finishes after a certain future moment.

3. Future Perfect Tense –
प्रयोग – इस Tense का प्रयोग तब किया जाता है जब कोई क्रिया भविष्य में किसी समय से पहले पूरी होनी हो; जैसे
(I will have finished my homework by evening.)
नियम – इस Tense में will / shall + have लगा कर Verb की तीसरी फार्म का प्रयोग किया जाता है।

4. Future Perfect Continuous Tense प्रयोग – इस Tense का प्रयोग तब किया जाता है यदि क्रिया भविष्य में किसी निश्चित समय (point of time) तक अथवा किसी निश्चित अवधिः (period of time) के लिए जारी रहनी हो; जैसे (By six o’clock I will have been sitting here for ten hours.)
नियम – इस Tense में will / shall + have been लगा कर Verb के ing वाले रूप का प्रयोग किया जाता है।

	Indefinite	Continuous	Perfect	Perfect Continuous
PRESENT PAST FUTURE	write (s) wrote will/shall write	I am/are writing was/were writing I will/shall be writing	has/have written had written will/shall have written	has/have been written had been written will/shall have been written

Miscellaneous Sentences For Practice

Fill in the blanks with the Present Indefinite form of the verbs given in the brackets :

1. He for a walk every day. (go)
2. She …………………….. up early in the morning. (get)
3. The peon ……………… the bell at 9 o’clock. (ring)
4. I ………………. my parents. (obey)
5. We ……………… our teachers. (respect)
6. We ……………………. our prayers in the morning.
7. She …………………….. sweetly. (sing
8. I do not …………………….. sweets. (like)

Fill in the blanks with the Present Continuous form of the verbs given in the brackets :

1. …………… it …………………… today? (rain)
2. ……….. the peon ……. the bell ? (ring)
3. ………… the boys …………..football ? (play)
4. He ……….. a picture. (paint)
5. We ………… our lesson. (learn)
6. She ……………………. not ……………. food. (cook)
7. I ……………………. a letter. (write)
8. You ………. (run)

Fill in the blanks with the Present Perfect form of the verbs given in the brackets :

1. I ….. my homework. (finish)
2. He …………………… first in the class. (stand)
3. …………you …………. your scooter ? (Sell)
4. ………….. she ……. a car? (buy)
5. They ……………………. their enemy. (murder)
6. We ……………………… the match. (win)
7. The peon ………………….. the bell. (ring)
8. We …….. our duty. (do)

Fill in the blanks with the Present Perfect Continuous form of the verbs given in the brackets :

1. It …………….. since morning. (rain)
2. She ………………….. from fever for a week. (suffer)
3. I ………………….. for you for half an hour. (wait)
4. He ……. his brother with money for five years. (help)
5. We ……………………. for a long time. (fight)
6. She ………………… for a job for several months. (look)
7. ……………………. he books from you since 2011? (borrow)
8. she ………………. in this office since 2013? (work)

Fill in the blanks with the Past Indefinite form of the verbs given in the brackets :

1. India ………………….. independence in 1947. (get)
2. He ………………….. government service in 2012. (join)
3. It …………………… heavily last night. (rain)
4. …………………… he …………… you money? (lend)
5. She ……………………. not ……………………. to my proposal. (agree)
6. Why ……………………. he ……………………. angry with you? (become)
7. The teacher ………………….. the student. (punish)
8. He …………………. to go for a walk daily. (use)

Fill in the blanks with the Past Continuous form of the verbs given in the brackets :

1. It ……………. in the morning. (rain)
2. The peon …… ……………. the bell. (ring)
3. Why …………….. she (weep)
4. She ……………………. clothes. (wash)
5. …………………. she …………………… food ? (eat)
6. …………………. the hunter ……………… birds ? (kill)
7. He a novel ………… (read)
8. My grandmother …………………… a story. (tell)

Fill in the blanks with the Past Perfect form of the verbs given in the brackets :

1. He ………………….. a tiger before I reached the forest. (kill)
2. She …………………… a sweater before I bought a new one. (knit)
3. I ……………….. money from my friend before I received my salary. (borrow)
4. The river ………………….. its banks before the dam was built. (overflow)
5. …………….. she …………… a new dress before his father came ? (buy)
6. Why …………………… he …………….. you before you offended him ? (abuse)
7. My uncle …………………… a new house before he retired from service. (build)
8. ……. you in the race before you defeated him in boxing ? (beat)

Fill in the blanks with the Past Perfect Continuous form of the verbs given in the brackets :

1. We in Delhi for ten years. (live)
2. He ……………….. in this office since 2011. (work)
3. She ……………… in this school for ten years. (teach)
4. ……………. your brother for five years ? (look after)
5. Who …………………. you with money all these years ? (help)
6. The farmer …….. the fields for many days. (plough)
7. The teacher …………………… the students for a long time. (guide)
8. They …….. their country for thirty years. (serve)

Fill in the blanks with Future Indefinite form of the verbs given in the brackets :

1. I ………………… to Delhi. (go)
2. We ……………………. you with money. (help)
3. The peon ………. the bell at 9 o’clock. (ring)
4. ………… she……………….. in time ? (come)
5. …………………… he …………………… you his book ? (lend)
6. She ……………………. a rich man. (marry)
7. The train ………………….. in time. (arrive)
8. They …………………… a match tomorrow. (play)

Fill in the blanks with the Future Continuous form of the verbs given in the brackets :

1. She ……………………. a picture. (paint)
2. I ………………….. a new car. (buy)
3. He ……………………. to Delhi tomorrow.
4. They ………………….. their friends. (help)
5. People ………………….. their work. (do)
6. ……………….. she ………………….. food ? (cook)
7. …………………… they …………………. in the river ? (bathe)
8. He ……………………. not ……………………. his friends. (cheat)

Fill in the blanks with the Future Perfect form of the verbs given in the brackets :

1. She …………………… her work in time. (finish)
2. He ………………….. you with money. (help)
3. They ………………… their clothes. (wash)
4. I ………………….. the situation. (handle)
5. She …………………… her promise. (break)
6. He ……………………. you money. (lend)
7. ……….. you English ? (teach)
8. ………………….. they ……………. bravely ? (fight)

Fill in the blanks with the Future Perfect Continuous form of the verbs given in the brackets :

1. It ………………… for two hours. (rain)
2. They …………….. here for many years. (live)
3. She …………………… English since 2008 (learn)
4. He …………………. Mathematics for many years. (teach)
5. …………………. she ………. in this office since 1985 ? (work)
6. He ……………………. you for a long time. (cheat)
7. They ………………….. the poor since 1982. (help)
8. …… she ……………….. food for the whole family for several (cook)

Concord : Subject-Verb Agreement

1. क्रिया का प्रयोग अवश्य ही इसके कर्ता के अनुसार होना चाहिए।
2. एकवचन कर्ता के साथ एकवचन क्रिया का प्रयोग किया जाता है और बहुवचन कर्ता के साथ बहुवचन क्रिया का प्रयोग किया जाता है।

Singular Subject – Singular Verb	Plural Subject – Plural Verb
This child is very clever. That book was interesting. He has many storybooks. He does not work regularly.	These children are very clever. Those books were interesting. They have many storybooks. They do not work regularly.

याद रखें – I के साथ am, was, have तथा you के साथ are, were, have का प्रयोग किया जाता है।

I am / was angry with him.
I have a good English dictionary.
You are / were late for school.
You have a good house.

3. निम्नलिखित क्रियाओं को negative और question form में बदलने के लिए have का प्रयोग किया जाता है, न कि has का –

do, does, did,
will, would, shall, should,
can, could, may, might,
must, need, dare,
ought to, used to
Examples :

• Does she have a new frock ?
• We do not have a high school in our village.
• Did he have his share ?
• I shall have a lot of money next week.
• He used to have a car.
• Mohan must have finished his work.

कुछ वाक्यों में ‘There’ कर्ता का स्थान ले लेता है। वास्तविक कर्ता क्रिया के बाद आता है।

• There is a table in this room.
• There are ten desks in this room.
• There was a clock on the wall.
• There were pictures on the wall.
• There has been a mistake.
• There have been many mistakes.

याद रखिए कि ऐसे वाक्यों में क्रिया वास्तविक कर्ता के अनुसार होती है। यदि वास्तविक कर्ता एकवचन (a table, a clock, a mistake) हो तो क्रिया भी एकवचन (is, was, has) होती है। यदि वास्तविक कर्ता बहुवचन (ten desks, pictures, many mistakes) हो तो क्रिया भी बहुवचन (are, were, have) होती है। परन्तु ‘There’ के विपरीत It’ के साथ सदा एकवचन क्रिया का प्रयोग किया जाता है।

• It is hot today.
• It was 7:30 at that time.
• It has been a mistake.

प्रायः हम एक भूल कर जाते हैं, जिसे “Error of Proximity” कहा जाता है, हम क्रिया को इसके साथ लगती संज्ञा की मात्रा के अनुसार कर देते हैं। निम्नलिखित उदाहरणों में दिखाए अनुसार हमें इनसे बचना चाहिए।

• The quality of the mangoes was (not ‘were) not good.
• His knowledge of Indian vernaculars is (not are) very sound.
• The state of his affairs was (not ‘were) such as to cause anxiety to his parents.
• The results of the new policy are (not is) very encouraging.

4. and’ द्वारा जोड़े गए दो या दो से अधिक एकवचन संज्ञाओं या सर्वनामों के साथ बहुवचन क्रिया का प्रयोग किया जाता है। जैसे –

• Gold and silver are precious metals.
• Fire and water do not agree.
• Are your father and mother at home ?
• He and I were playing.

परन्तु यदि संज्ञाएं हमारे दिमाग को एक विचार देती हैं, अथवा किसी एक ही व्यक्ति अथवा वस्तु की ओर संकेत करती हैं, तो एकवचन क्रिया का प्रयोग किया जाता है। जैसे –

• The horse and carriage is at the door.
• Bread and butter is his only food.
• Honour and glory is his reward.
• The novelist and poet is dead.

5. एकवचन के साथ with, as well as, आदि द्वारा जोड़े गए शब्द उपवाक्य सम्बन्धी होते हैं। इसलिए एकवचन क्रिया का प्रयोग किया जाना चाहिए; जैसे –

• The house, with its contents, was insured.
• The Chairman, with the directors, is to be present.
• The ship, with its crew, was lost.
• Silver, as well as cotton, has fallen in price.
• Sanskrit, as well as Arabic, was taught there.
• Justice, as well as mercy, allows it.

6. or अथवा nor द्वारा जोड़े गए दो या दो से अधिक कर्ताओं के साथ एकवचन क्रिया का प्रयोग किया जाता है। जैसे –

• No nook or corner was left unexplored.
• Our happiness or our sorrow is largely due to our own actions.
• Either the cat or the dog has been here.
• Neither food nor water was to be found there.
• Neither praise nor blame seems to affect him.

7. जब or अथवा nor द्वारा जोड़े गए कर्ताओं में से एक बहुवचन हो, तब क्रिया भी बहुवचन होनी चाहिए, और बहुवचन कर्ता को क्रिया के समीपतम् रखना चाहिए; जैसे –
Neither the chairman nor the directors are present. जब or अथवा nor द्वारा जोड़े गए कर्ता अलग-अलग सर्वनामों से सम्बन्धित हों तो क्रिया समीपतम् सर्वनाम के अनुसार होनी चाहिए; जैसे –

• Either he or I am mistaken
• Either you or he is mistaken.
• Neither you nor he is to blame.
• Neither my friend nor I am to blame.

Either, neither, each, everyone, many a के साथ एकवचन क्रिया का प्रयोग किया जाना चाहिए; जैसे

• He asked me whether either of the applicants was suitable.
• Neither of the two men was very strong.
• Each of these minerals is found in India.
• Everyone of the boys loves to ride.
• Many a man has done so.

ऐसी दो संज्ञाएं जिनसे पूर्व each अथवा every लगा हो और वे and द्वारा जुड़ी हुई हों, तो ऐसी स्थिति में उनके साथ एकवचन क्रिया का प्रयोग किया जाता है जैसे –

• Each boy and each girl was present in the class.
• Every table and every chair has been dusted.

8. कुछ संज्ञाएं जिनका रूप बहुवचन में होता है, परन्तु जिनका अर्थ एकवचन में होता है, ऐसी संज्ञाओं के साथ एकवचन क्रिया का प्रयोग किया जाता है; जैसे –

• The news is true.
• Politics was with him the business of his life.
• The wages of sin is death.
• Mathematics is a branch of study in every school.

कुछ संज्ञाएं जिनका रूप एकवचन में होता है, परन्तु जिनका अर्थ बहुवचन में होता है, ऐसी संज्ञाओं के साथ बहुवचन क्रिया का प्रयोग किया जाता है; जैसे – Twelve dozen cost one hundred rupees.

9. किसी समूह-वाचक संज्ञा के साथ एकवचन क्रिया का प्रयोग होता है जब उस समूह को समूचे पूर्णांक के रूप में ले लिया जाए; और बहुवचन क्रिया का प्रयोग तब होता है जब उसी समूह को उसकी विभिन्न इकाइयों के रूप में लिया जाए; जैसे –
1. The committee has issued its report.
2. The committee are divided on one minor point.
परन्तु हमें समरूप होना चाहिए। इस प्रकार, हमें कहना चाहिए :
The committee has appended a note to its (not their) report.

10. जब कोई बहुवचन संज्ञा किसी एक वस्तु अथवा किसी सामूहिक इकाई का एक विशेष नाम हो, तब इसके साथ एकवचन क्रिया का प्रयोग किया जाना चाहिए; जैसे –

• The Arabian Nights is still a great favourite.
• The United States has a big navy.
• Plutarch’s Lives is an interesting book.
• Gulliver’s Travels was written by Swift.

11. जब कोई बहुवचन संज्ञा किसी ऐसी विशेष संख्या अथवा मात्रा की ओर संकेत करती है जिसे एक समूचा पूर्णांक माना गया हो, तब क्रिया प्रायः एकवचन होती है; जैसे –

• One hundred paise is equal to one rupee.
• Six miles is a long distance.
• Fifty thousand rupees is a large sum.
• Three parts of the business is left for me to do.

Complete these sentences with one of the two verbs given in brackets :

1. They ……….. going to report the matter to the Principal. (is / are)
2. She ………… not have any chance of winning the election. (do / does)
3. The teacher does not ……….. much work to do. (has / have)
4. ……….. you have any money with you ? (Do / Does)
5. Doesn’t Chennai ……….. a severe winter ? (has / have)
6. ………… the men shouting loudly ? (Was / Were)
7. Can we ………… coffee instead of tea ? (has / have)
8. Don’t the students ……………. their books with them ? (has / have)
9. ……………. your friend visit you daily ? (Do / Does)
10. They needn’t …………… gone there so early. (has / have)
Answer:
1. are
2. does
3. have
4. Do
5. have
6. Were
7. have
8. have
9. Does
10. have.

Choose the correct verb out of the two given in brackets and rewrite the sentences :

1. The cause of both the accidents (was, were) the same.
2. Each of the boys (has, have) done his homework.
3. All my friends (has, have) joined this college.
4. (Have, has) either of you any money?
5. Everyone of us (is, are) going to the film.
6. Both the rooms (was, were) whitewashed…
7. One of the debators (was, were) very interesting.
8. The furniture (is, are) to be sold today.
9. A few sheep (was, were) running about in the field.
10. Neither Manu nor Misha (was, were) with us today.
Answer:
1. was
2. has
3. have
4. Has
5. is
6. were
7. was
8. is
9. were
10. was.

Sequence Of Tenses

यदि मुख्य-वाक्य की क्रिया Present Tense में हो, तो उपवाक्य की क्रिया किसी भी Tense में हो सकती है; जैसे

1. He assures me that the train has arrived.
2. He says she will come at 4 p.m.
3. The Principal declares that latecomers will be fined.
4. Subhash says that they had been sitting there for two hours.
5. The witness affirms that the accused is not guilty.

2. यदि मुख्य वाक्य की क्रिया Future Tense में हो, तो उप-वाक्य की क्रिया किसी भी Tense में हो सकती है; जैसे
1. He will assure me that the train has arrived.
2. He will assure me that the train had arrived.
3. He will say that Ramesh is late.
4. He will say that Ramesh was late.
5. He will tell us that Radha is waiting for us at the bus stop.

3. यदि मुख्य-वाक्य की क्रिया Past Tense में हो, तो उप-वाक्य की क्रिया भी Past Tense में ही होनी चाहिए। (किन्तु यदि उप-वाक्य में किसी Habit या Universal truth का वर्णन हो, तो इसकी क्रिया Present Tense में ही रहती है।)
1. He assured me that the train had arrived.
2. She said that she would come at 4 p.m.
3. The Principal declared that latecomers would be fined.
4. He told me that he never tells a lie.
5. The teacher said that honesty is the best policy.

If Patterns

(If से आरम्भ होने वाले उप-वाक्य) नीचे दिए गए चार्ट को सदा ध्यान में रखिए- ‘.

If Clause	Main Clause
I   Form II   Form Had + III Form	will + I Form would + I Form would have + III Form

नोट : Until और unless से आरम्भ होने वाले उप-वाक्यों में भी यही नियम लागू होते हैं।
Examples :
1. If you play the piano, I will sing.
If you played the piano, I would sing.
If you had played the piano, I would have sung.

2. He will beat the thief if he catches him.
He would beat the thief if he caught him.
He would have beaten the thief if he had caught him.

3. The horse will kick you if you go near it.
The horse would kick you if you went near it.
The horse would have kicked you if you had gone near it.

4. He will not play tennis if you do not give him a racket.
He would not play tennis if you did not give him a racket.
He would not have played tennis if you had not given him a racket.

5. He will walk to the station if he is well.
He would walk to the station if he was well.
He would have walked to the station if he had been well.

6. My servant can paint your boat if you give him a brush.
My servant could paint your boat if you gave him a brush.
My servant could have painted your boat if you had given him a brush.

Punjab State Board PSEB 9th Class Maths Book Solutions Chapter 9 Areas of Parallelograms and Triangles Ex 9.1 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 9 Maths Chapter 9 Areas of Parallelograms and Triangles Ex 9.1

Question 1.
Which of the following figures lie on the same base and between the same parallels. In such a case, write the common base and the two parallels.

(i)

Answer:
In figure (i), trapezium ABCD and ∆ PDC lie on the same base and between the same parallels.
Here, DC is the common base and DC and AB are two parallels.

(ii)

Answer:
In figure (ii), no two figures lie on the same base and between the same parallels.

(iii)

Answer:
In figure (iii), parallelogram PQRS and ∆ TQR lie on the same base and between the same parallels.
Here, QR is the common base and QR and PS are two parallels.

(iv)

Answer:
In figure (iv), no two figures lie on the same base and between the same parallels.

(v)

Answer:
In figure (v), parallelograms ABCD and APQD as well as trapeziums ABQD and APCD, all the four figures, lie on the same base and between the same parallels.
Here, AD is the common base and AD and BQ are two parallels.

(vi)

Answer:
In figure (vi), no two figure lie on the same base and between the same parallels.

Punjab State Board PSEB 9th Class Maths Book Solutions Chapter 8 Quadrilaterals MCQ Questions with Answers.

PSEB 9th Class Maths Chapter 8 Quadrilaterals MCQ Questions

Multiple Choice Questions and Answer

Answer each question by selecting the proper alternative from those given below each question to make the statement true:

Question 1.
The ratio of four angles in order of a quadrilateral is 2 : 4 : 5 : 4. Then, the measure of the smallest angle of the quadrilateral is
A. 120°
B. 96°
C. 48°
D. 60°
Answer:
C. 48°

Question 2.
In quadrilateral PQRS, ∠P = 5x, ∠Q = 3x, ∠R = 4x and ∠S = 6x. Then, the measure of the greatest angle of quadrilateral PQRS is …………… .
A. 100°
B. 60°
C. 80°
D. 120°
Answer:
D. 120°

Question 3.
In quadrilateral ABCD, ∠A + ∠B = 150°.
Then ∠C + ∠D =
A. 105°
B. 210°
C. 150°
D. 300°
Answer:
B. 210°

Question 4.
In trapezium PQRS, PQ || RS. If ∠P = 150°, then ∠S = …………. .
A. 75°
B. 150°
C. 60°
D. 30°
Answer:
D. 30°

Question 5.
The perimeter of parallelogram ABCD is 22 cm.
If AB = 4 cm, then BC = ……………. cm.
A. 7
B. 6
C. 5.5
D. 4
Answer:
A. 7

Question 6.
In parallelogram ABCD, ∠A – ∠B = 30°. Then, ∠C = ……………… .
A. 105°
B. 75°
C. 150°
D. 60°
Answer:
A. 105°

Question 7.
In parallelogram ABCD, the bisectors of ∠A and ∠B intersect at M. If ∠A = 80°, then ∠AMB = ……………. .
A. 40°
B. 50°
C. 80°
D. 90°
Answer:
D. 90°

Question 8.
In parallelogram ABCD, the ratio ∠A : ∠B : ∠C : ∠D can be
A. 3 : 4 : 5 : 6
B. 2 : 3 : 3 : 2
C. 2 : 3 : 2 : 3
D. 2 : 3 : 5 : 8
Answer:
C. 2 : 3 : 2 : 3

Question 9.
In parallelogram ABCD, 3 ∠ A = 2 ∠ B. Then, ∠ D = ………………. .
A. 120°
B. 108°
C. 72°
D. 60°
Answer:
B. 108°

Question 10.
In ∆ ABC, E and F are the midpoints of AB and AC respectively. If EF = 4 cm, then BC = …………… cm.
A. 8
B. 2
C. 4
D. 12
Answer:
A. 8

Question 11.
In ∆ ABC, P is the midpoint of AB and Q is the midpoint of AC. Then, PQCB is a ………….. .
A. trapezium
B. parallelogram
C. rectangle
D. rhombus
Answer:
A. trapezium

Question 12.
In ∆ ABC, D, E and F are the midpoints of AB, BC and CA respectively. If the perimeter of ∆ DEF is 30 cm, then the perimeter of ∆ ABC is ……………. cm.
A. 15
B. 30
C. 45
D. 60
Answer:
D. 60

Question 13.
∆ ABC is an equilateral triangle. D, E and F are the midpoints of AB, BC and CA respectively. If AB = 8 cm, the perimeter of ∆ DEF is …………… cm.
A. 24
B. 12
C. 6
D. 48
Answer:
B. 12

Question 14.
ABCD is a rectangle. If AB = 5 cm and BC = 12
cm, then BD = ………………. cm
A. 17
B. 13
C. 8.5
D. 1
Answer:
B. 13

Question 15.
ABCD is a rhombus. If AC = 10 cm and BD = 24 cm, the perimeter of ABCD is …………………. cm.
A. 13
B. 26
C. 52
D. 48
Answer:
C. 52

Punjab State Board PSEB 9th Class Maths Book Solutions Chapter 8 Quadrilaterals Ex 8.2 Textbook Exercise Questions and Answers.

PSEB Solutions for Class 9 Maths Chapter 8 Quadrilaterals Ex 8.2

Question 1.
ABCD is a quadrilateral in which F Q, R and S are midpoints of the sides AB, BC, CD and DA respectively (see the given figure 1). AC is a diagonal. Show that:
(i) SR || AC and SR = \(\frac{1}{2}\) AC
(ii) PQ = SR
(iii) PQRS is a parallelogram.

Answer:
In ∆ DAC, S and R are the midpoints of DA and DC respectively.
Through C draw a line parallel to AD which intersects line SR at T.
In ∆ DRS and ∆ CRT
∠ DRS = ∠ CRT (Vertically opposite angles)
∠ RSD = ∠ RTC (Alternate angles formed by transversal ST of DS || TC)
DR = CR (R is the midpoint of DC.)
∴ ∆ DRS ≅ ∆ CRT (AAS rule)
∴ DS = CT and SR = RT (CPCT)
As S is the midpoint of DA, we have DS = SA.
∴ SA = CT
And, by construction, SA || CT.
∴ Quadrilateral SACT is a parallelogram.
∴ ST || AC
∴ SR || AC ………… (1)
Now, SR = RT gives SR = \(\frac{1}{2}\)ST
In parallelogram SACT, ST = AC.
∴ SR = \(\frac{1}{2}\)AC ……………. (2)
Taking (1) and (2) together,
SR || AC and SR = \(\frac{1}{2}\)AC ….. Result (1)
Similarly, in ∆ ABC, P and Q are the midpoints of AB and BC respectively. ,
∴ PQ || AC and PQ = \(\frac{1}{2}\)AC
Now, SR = \(\frac{1}{2}\)AC and PQ = \(\frac{1}{2}\)AC
∴ PQ = SR …… Result (ii)
Similarly, SR || AC and PQ || AC.
∴ PQ || SR
Thus, in quadrilateral PQRS, PQ = SR and PQ || SR.
Hence, by theorem 8.8, PQRS is a parallelogram. … Result (iii)

Question 2.
ABCD is a rhombus and F Q, R and S are the midpoints of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.
Answer:

ABCD is a rhombus and F Q, R and S are the midpoints of sides AB, BC, CD and DA respectively.
∴ In ∆ ABC, PQ || AC and PQ = \(\frac{1}{2}\)AC.
∴ In ∆ ADC, SR || AC and SR = \(\frac{1}{2}\)AC.
Hence, in quadrilateral PQRS, PQ || SR and PQ = SR.
∴ Quadrilateral PQRS is a parallelogram.
Now, since ABCD is a rhombus, AC and BD bisect each other at right angles at M.
∴ ∠ AMB = 90°
Now, AC || PQ and MN is their transversal.
∴ ∠ AMN + ∠ MNP = 180° (Interior angles on the same side of transversal)
∴ ∠ AMB + ∠MNP = 180°
∴ 90° + ∠ MNP = 180°
∴ ∠ MNP = 90°
In ∆ ABD, P and S are the midpoints of AB and AD respectively.
∴ PS || BD and NP is their transversal.
∴ ∠ DNP + ∠ NPS = 180°
∴ ∠ MNP + ∠ NPS =180°
∴ 90° + ∠ NPS = 180°
∴ ∠ NPS = 90°
∴ ∠ SPQ = 90°
Thus, in parallelogram PQRS, one angle ∠P is a right angle.
Hence, quadrilateral PQRS is a rectangle.

Question 3.
ABCD is a rectangle and P, Q, R and S are midpoints of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.
Answer:

Since ABCD is a rectangle, its diagonals are equal.
∴ AC = BD
∴ \(\frac{1}{2}\)AC = \(\frac{1}{2}\)BD
In ∆ ABC, P and Q are the midpoints of AB and BC respectively.
∴ PQ = \(\frac{1}{2}\)AC
Similarly, in ∆ ADC, SR = \(\frac{1}{2}\)AC; in ∆ ABD, SP = \(\frac{1}{2}\) BD and in ∆ BCD, QR = \(\frac{1}{2}\) BD.
Now, PQ = SR = \(\frac{1}{2}\)AC, SP = QR = \(\frac{1}{2}\)BD and \(\frac{1}{2}\)AC = \(\frac{1}{2}\)BD
Hence, in quadrilateral PQRS,
PQ = QR = RS = SP
Thus, all the sides of quadrilateral PQRS are equal.
Hence, quadrilateral PQRS is a rhombus.

Question 4.
ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the midpoint of AD. A line is drawn through E parallel to AB intersecting BC at F (see the given figure). Show that F is the midpoint of BC.

Answer:
Suppose line EF drawn through E and parallel to AB intersects BD at M.
EF || AB and AB || DC
∴ EF || DC
Trapezium ABCD is divided into two triangles, ∆ ABD and ∆ BCD, by diagonal BD.
In ∆ ABD, E is the midpoint of AD and a line through E and parallel to AB intersects BD at M.
Hence, by theorem 8.10, M is the midpoint of BD.
Now, in ∆ BCD, M is the midpoint of BD and a line through M and parallel to CD intersects BC at F.
Hence, by theorem 8.10, F is the midpoint of BC.
Note: The following result about the length of EF can also be derived:
EF = \(\frac{1}{2}\)(AB + CD)
Moreover, if X and Y are the midpoints of the diagonals of above trapezium ABCD, then XY = \(\frac{1}{2}\)|AB – CD|.

Question 5.
In a parallelogram ABCD, E and F are the midpoints of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.

Answer:
E and F are the midpoints of AB and CD respectively.
∴ AE = \(\frac{1}{2}\)AB and CF = \(\frac{1}{2}\)CD
In parallelogram ABCD, AB = CD and AB || CD.
∴ AE = CF and AE || CF
Hence, quadrilateral AECF is a parallelogram.
∴ AF || EC
∴ AP || EQ
In ∆ ABP E is the midpoint of AB and EQ || AR
∴ Q is the midpoint of PB. (Theorem 8.10)
∴PQ = QB …………… (1)
Similarly, in ∆ DQC, F is the midpoint of DC and FP || CQ.
∴ P is the midpoint of DQ. (Theorem 8.10)
∴ DP = PQ …………….. (2)
From (1) and (2), DP = PQ = QB.
Moreover, DP + PQ + QB = BD.
Thus, AF and EC trisect the diagonal BD.

Question 6.
Show that the line segments joining the midpoints of the opposite sides of a quadrilateral bisect each other.
Answer:

In quadrilateral ABCD, P Q, R and S are the midpoints of sides AB, BC, CD and DA respectively.
In ∆ ABC, P and Q are the midpoints of AB and BC respectively.
∴ PQ || AC and PQ = \(\frac{1}{2}\)AC …………….. (1)
In ∆ ADC, S and R are the midpoints of DA and DC respectively.
∴ SR || AC and SR = \(\frac{1}{2}\)AC ……………… (2)
From (1) and (2),
PQ = SR and PQ || SR.
Thus, in quadrilateral PQRS, sides in one pair of opposite sides are equal and parallel. Hence, quadrilateral PQRS is a parallelogram. The diagonals of a parallelogram bisect each other. [Theorem 8.6]
∴ PR and SQ bisect each other.
Thus, the line segments joining the midpoints of the opposite sides of a quadrilateral bisect each other.

Question 7.
ABC is a triangle right angled at C. A line through the midpoint M of hypotenuse AB and parallel to BC intersects AC at D. Show that:
(i) D is the midpoint of AC.
(ii) MD ⊥ AC
(iii) CM = MA = \(\frac{1}{2}\)AB
Answer:

In ∆ ABC, ∠ C is a right angle and M is the midpoint of hypotenuse AB. A line through M and parallel to BC intersects AC at D.
Hence, by theorem 8.10, DM bisects AC.
∴ D is the midpoint of AC. ….. Result (i)
In ∆ ABC, ∠ C is a right angle.
∴ ∠ C = 90°
Now, BC || DM and DC is their transversal.
∴ ∠ MDC + ∠ DCB = 180° (Interior angles on the same side of transversal)
∴ ∠ MDC + 90° = 180°
∴ ∠ MDC = 90°
Thus, MD is perpendicular to AC.
∴ MD ⊥ AC …… Result (ii)
Now, in ∆ ADM and ∆ CDM,
AD = CD (D is the midpoint of AC)
∠ ADM = ∠ CDM (Right angles)
DM = DM (Common)
∴ ∆ ADM ≅ ∆ CDM (SAS rule)
∴ AM = CM (CPCT) ……………. (1)
Now, M is the midpoint of AB.
∴ AM = \(\frac{1}{2}\)AB …… (2)
< Prom (1) and (2),
CM = MA = \(\frac{1}{2}\)AB …… Result (iii)