# NCERT Solutions for Class 9 Maths Exercise 7.3

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NCERT solutions for Class 9 Maths Triangles

## NCERT Solutions for Class 9 Mathematics Triangles

###### 1. ABC and DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (See figure). If AD is extended to intersect BC at P, show that:

(i) ABD ACD

(ii) ABP ACP

(iii) AP bisects A as well as D.

(iv) AP is the perpendicular bisector of BC.

Ans. (i) ABC is an isosceles triangle.

AB = AC

DBC is an isosceles triangle.

BD = CD

Now in ABD and ACD,

AB = AC [Given]

BD = CD [Given]

ABD ACD [By SSS congruency]

(ii) Now in ABP and ACP,

AB = AC [Given]

AP = AP

ABP ACP [By SAS congruency]

(iii) Since ABP ACP [From part (ii)]

BAP = CAP [By C.P.C.T.]

AP bisects A.

Since ABD ACD [From part (i)]

Now ADB + BDP = [Linear pair] ……….(iii)

And ADC + CDP = [Linear pair] ……….(iv)

From eq. (iii) and (iv),

BDP = CDP

DP bisects D or AP bisects D.

(iv) Since ABP ACP [From part (ii)]

BP = PC [By C.P.C.T.] ……….(v)

And APB = APC [By C.P.C.T.] …….(vi)

Now APB + APC = [Linear pair]

APB + APC = [Using eq. (vi)]

2APB =

APB =

AP BC ……….(vii)

From eq. (v), we have BP PC and from (vii), we have proved AP B. So, collectively AP is perpendicular bisector of BC.

NCERT Solutions for Class 9 Maths Exercise 7.3

###### 2. AD is an altitude of an isosceles triangle ABC in which AB = AC. Show that:

Ans. In ABD and ACD,

AB = AC [Given]

ABD ACD [RHS rule of congruency]

BD = DC [By C.P.C.T.]

NCERT Solutions for Class 9 Maths Exercise 7.3

###### 3. Two sides AB and BC and median AM of the triangle ABC are respectively equal to side PQ and QR and median PN of PQR (See figure). Show that:

(i) ABM PQN

(ii) ABC PQR

Ans. AM is the median of ABC.

BM = MC = BC ……….(i)

PN is the median of PQR.

QN = NR = QR ……….(ii)

Now BC = QR [Given] BC = QR

BM = QN ……….(iii)

(i) Now in ABM and PQN,

AB = PQ [Given]

AM = PN [Given]

BM = QN [From eq. (iii)]

ABM PQN [By SSS congruency]

B = Q [By C.P.C.T.] ……….(iv)

(ii) In ABC and PQR,

AB = PQ [Given]

B = Q [Prove above]

BC = QR [Given]

ABC PQR [By SAS congruency]

NCERT Solutions for Class 9 Maths Exercise 7.3

###### 4. BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

Ans. In BEC and CFB,

BEC = CFB [Each ]

BC = BC [Common]

BE = CF [Given]

BEC CFB [RHS congruency]

EC = FB [By C.P.C.T.] …..(i)

Now In AEB and AFC

AEB = AFC [Each ]

A = A [Common]

BE = CF [Given]

AEB AFC [ASA congruency]

AE = AF [By C.P.C.T.] …………(ii)

Adding eq. (i) and (ii), we get,

EC + AE = FB + AF

AB = AC

ABC is an isosceles triangle.

NCERT Solutions for Class 9 Maths Exercise 7.3

###### 5. ABC is an isosceles triangles with AB = AC. Draw AP BC and show that B = C.

Ans. Given: ABC is an isosceles triangle in which AB = AC

To prove: B = C

Construction: Draw AP BC

Proof: In ABP and ACP

APB = APC = [By construction]

AB = AC [Given]

AP = AP [Common]

ABP ACP [RHS congruency]

B = C [By C.P.C.T.]

## NCERT Solutions for Class 9 Maths Exercise 7.3

NCERT Solutions for Class 9 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 9 Maths includes text book solutions from Mathematics Book. NCERT Solutions for CBSE Class 9 Maths have total 15 chapters. 9 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 9 solutions PDF and Maths ncert class 9 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide.

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