4. In the given figure, two sides AB and BC and the median AD of ÎA

1.	4. In the given figure, two sides AB and BC and the median AD of ÎABC areequal respectively to the two sides PQ and QR and the median PM of theother triangle PQR.INCERTProve that[CBSE 2011]
Answer» Solution: Given: AD is the median of ∆ABC & PM is the median of ∆PQR. AB = PQ, BC = QR & AD = PM To Show: (i) ΔABD ≅ ΔPQM(ii) ΔABC ≅ ΔPQR Proof: Since AD & PM is the median of ∆ABC (i)1/2 BC = BD & 1/2QR = QM (AD and PM are median) Now, BC = QR. (given) ⇒ 1/2 BC = 1/2QR (Divide both sides by 2) ⇒ BD = QM In ΔABD and ΔPQM, AD = PM (Given) AB = PQ (Given) BD = QM (Proved above) Therefore, ΔABD ≅ ΔPQM (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)

4. In the given figure, two sides AB and BC and the median AD of ÎABC areequal respectively to the two sides PQ and QR and the median PM of theother triangle PQR.INCERTProve that[CBSE 2011]

Answer»

Solution:

Given: AD is the median of ∆ABC & PM is the median of ∆PQR.

AB = PQ, BC = QR & AD = PM

To Show: (i) ΔABD ≅ ΔPQM(ii) ΔABC ≅ ΔPQR

Proof: Since AD & PM is the median of ∆ABC

(i)1/2 BC = BD & 1/2QR = QM (AD and PM are median) Now, BC = QR. (given)

⇒ 1/2 BC = 1/2QR (Divide both sides by 2) ⇒ BD = QM

In ΔABD and ΔPQM, AD = PM (Given) AB = PQ (Given) BD = QM (Proved above)

Therefore, ΔABD ≅ ΔPQM (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)