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CBSE Class 9 Mathematics Question 19 of 21

Triangles — Question 3

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Question

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 Δ PQR. Show that :

(i) Δ ABM ≅ Δ PQN

(ii) Δ ABC ≅ Δ PQR

Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of Δ PQR. Show that : NCERT Class 9 Mathematics CBSE Solutions.
Two sides AB and BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of Δ PQR. Show that : NCERT Class 9 Mathematics CBSE Solutions.
Answer

Given :

AB = PQ, BC = QR = x (let) and AM = PN.

(i) Given,

AM is the median of △ ABC

∴ BM = CM = 12BC=x2\dfrac{1}{2}BC = \dfrac{x}{2} .....(1)

Also, PN is the median of △ PQR

∴ QN = RN = 12QR=x2\dfrac{1}{2}QR = \dfrac{x}{2} ......(2)

From equation (1) and (2), we get :

BM = QN ..........(3)

Now, in △ ABM and △ PQN we have,

⇒ AB = PQ (Given)

⇒ BM = QN [From equation (3)]

⇒ AM = PN (Given)

∴ △ ABM ≅ △ PQN (By S.S.S. congruence rule)

Hence, proved that △ ABM ≅ △ PQN.

(ii) Since,

△ ABM ≅ △ PQN

We know that,

Corresponding parts of the congruent triangle are equal.

∠B = ∠Q (By C.P.C.T.) ...........(4)

Now, In △ ABC and △ PQR we have

⇒ AB = PQ (Given)

⇒ ∠B = ∠Q [From equation (4)]

⇒ BC = QR (Given)

∴ △ ABC ≅ △ PQR (By S.A.S. congruence rule)

Hence, proved that △ ABC ≅ △ PQR.