# RD Sharma Solutions Class 9 Congruent Triangles Exercise 10.4

## RD Sharma Solutions Class 9 Chapter 10 Ex 10.4

(1) In fig (10).9(2) It is given that AB = CD and AD = BC. Prove that $\Delta ADC \cong \Delta CBA$.

Solution:

Given that in the figure AB = CD  and AD = BC.

We have to prove $\Delta ADC \cong \Delta CBA$

Now,

Consider $\Delta$ ADC and $\Delta$ CBA.

We have

AB = CD                     [Given]

And AC = AC             [Common side]

So, by SSS congruence criterion, we have

$\Delta ADC \cong \Delta CBA$

Hence proved

(2) In a $\Delta$ PQR. IF PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.

Sol: Given that in $\Delta$ PQR, PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively

We have to prove LN = MN.

Join L and M, M and N, N and L

We have PL = LQ, QM = MR and RN = NP

[Since, L, M and N are mid-points of Pp. QR and RP respectively]

And also PQ = QR

• PL = LQ = QM = MR = $\frac{PQ}{2}$ = $\frac{QR}{2}$ ……(i) Using mid-point theorem,

We have

MN $\parallel$ PQ and MN = $\frac{PQ}{2}$

• MN = PL = LQ ……(ii)

Similarly, we have

LN $\parallel$  QR and LN = (1/2)QR

• LN = QM = MR ……(iii)

From equation (i), (ii) and (iii), we have

PL = LQ = QM = MR = MN = LN

LN = MN

#### Practise This Question

Which of the following is not a binary operation defined on the set of positive real numbers?