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# Through A, B and C, lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a ∆ ABC as shown in Fig.8.5. Show that BC = 1/2 QR.

**Solution:**

Given, ABC is a triangle

Lines RQ, PR and QP are drawn through A, B and C __parallel__ to sides BC, CA and AB of the triangle ABC.

We have to show that BC = 1/2 QR

Given, RQ || BC

PR || AC

QP || AB

Considering __quadrilateral__ BCAR,

BR || CA

RA || BC

We know the opposite sides of a __parallelogram__ are parallel and __congruent__.

So, BCAR is a parallelogram.

BC = AR ------------------ (1)

Considering quadrilateral BCQA,

BC || AQ

AB || QC

So, BCQA is a parallelogram

BC = AQ ------------------ (2)

Adding (1) and (2),

BC + BC = AR + AQ

2BC = AR + AQ

From the figure,

AR + AQ = RQ

So, 2BC = RQ

Therefore, BC = 1/2 QR

**✦ Try This: **In parallelogram ABCD, two point P and Q are taken on diagonal BD such that DP = BQ. Show that △AQB ≅ △CPD

**☛ Also Check:** NCERT Solutions for Class 9 Maths Chapter 8

**NCERT Exemplar Class 9 Maths Exercise 8.3 Problem 7**

## Through A, B and C, lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a ∆ ABC as shown in Fig.8.5. Show that BC = 1/2 QR.

**Summary:**

Through A, B and C, lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a ∆ ABC as shown in Fig.8.5. It is shown that BC = 1/2 QR

**☛ Related Questions:**

- D, E and F are the mid-points of the sides BC, CA and AB, respectively of an equilateral triangle AB . . . .
- Points P and Q have been taken on opposite sides AB and CD, respectively of a parallelogram ABCD suc . . . .
- In Fig. 8.7, P is the mid-point of side BC of a parallelogram ABCD such that ∠BAP = ∠DAP. Prove that . . . .

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