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A △ ABC is given. If lines are drawn through A, B, C, parallel respectively to the sides BC, CA and AB, forming △ PQR, as shown in the adjoining figure, show that BC = ½ QR.

Answer:

We know that AR || BC and AB || RC

From the figure we know that ABCR is a parallelogram

So we get

AR = BC …… (1)

We know that AQ || BC and QB || AC

From the figure we know that AQBC is a parallelogram

So we get

QA = BC ……… (2)

By adding both the equations

AR + QA = BC + BC

We know that AR + QA = QR

So we get

QR = 2BC

Dividing by 2

BC = QR/2

BC = ½ QR

Therefore, it is proved that BC = ½ QR.

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