Two sides ab and bc and median am

Asked by admin @ 25/08/2021 in Math viewed by 350 People

"Question 3 Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of ΔPQR (see the given figure). Show that: (i) ΔABM ≅ ΔPQN (ii) ΔABC ≅ ΔPQR

Class 9 - Math - Triangles Page 128"

Answered by admin @ 25/08/2021

Congruence of triangles:

Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

Criteria for congruence of triangles:

There are 4 criteria for congruence of triangles.

SAS( side angle side):

Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.

SSS(side side side):

Three sides of One triangle are equal to the three sides of another triangle then the two Triangles are congruent.

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We know that median bisects opposite side. Use this property and then show that given parts by using SSS and SAS congruence rule.

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Solution:

Given:

AM is the median of ∆ABC & PN is the median of ∆PQR.

AB = PQ, BC = QR & AM = PN

To Show:

(i) ΔABM ≅ ΔPQN
(ii) ΔABC ≅ ΔPQR

Proof:

Since AM & PN is the median of ∆ABC

(i) 1/2 BC = BM &

1/2QR = QN

(AM and PN are median)

Now,

BC = QR. (given)

⇒ 1/2 BC = 1/2QR

(Divide both sides by 2)

⇒ BM = QN

In ΔABM and ΔPQN,

AM = PN (Given)

AB = PQ (Given)

BM = QN (Proved above)

Therefore,

ΔABM ≅ ΔPQN

(by SSS congruence rule)

∠B = ∠Q (CPCT)

(ii) In ΔABC & ΔPQR,

AB = PQ (Given)

∠B = ∠Q(proved above in part i)

BC = QR (Given)

Therefore,

ΔABC ≅ ΔPQR

( by SAS congruence rule)

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Hope this will help you......

Two sides ab and bc and median am

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