In the figure- two sides A B- B C and the median A D of - A B C are respectively equal to two sides P Q- Q R and median P S of - P Q R- Prove that i A D B - P S Q-ii A D C - P S R-Does it follow that triangles A B C and P Q R are congruent-

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Byju's Answer

Standard VIII

Mathematics

SAS Postulate

In the figure...

Question

In the figure, two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR. Prove that

(i) ΔADB ≅ ΔPSQ;

(ii) ΔADC ≅ ΔPSR.

Does it follow that triangles ABC and PQR are congruent?

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Solution

Given: AB = PQ, BC = QR and AD = PS

Also, BD = DC and QS = SR

To Prove: i) ΔADB ≅ ΔPSQ and ii) ΔADC ≅ ΔPSR

Proof:

BC = QR (Given)

BD + DC = QS + SR

2DC = 2SR

DC = SR

or, 2BD = 2QS

BD = QS

(i) In ΔADB and ΔPSQ:

BD = QS

AB = PQ (Given)

AD = PS (Given)

∴ΔADB ≅ ΔPSQ … (1) (SSS congruency)

(ii) In ΔADC and ΔPSR:

DC = SR

AD = PS (Given)

∴ΔADC ≅ ΔPSR … (2) (SAS congruency)

Also, from (1) and (2), ΔABC ≅ ΔPQR

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In the figure, two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR. Prove that (i) ΔADB ≅ ΔPSQ; (ii) ΔADC ≅ ΔPSR. Does it follow that triangles ABC and PQR are congruent?

In the figure, two sides AB, BC and the median AD of ΔABC are respectively equal to two sides PQ, QR and median PS of ΔPQR. Prove that

(i) ΔADB ≅ ΔPSQ;

(ii) ΔADC ≅ ΔPSR.

Does it follow that triangles ABC and PQR are congruent?