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Quadrilaterals — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsQuadrilaterals5 Marks
Question
A $\triangle\text{ABC}$ is given. If lines are drawn through A, B, C, parallel respectively to the sides BC, CA and AB, forming $\triangle\text{PQR},$ as shown in the adjoining figure, show that $\text{BC}=\frac{1}{2}\text{QR}.$

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Answer

Given: A $\triangle\text{ABC}$ in which through points A, B and C, line QR, QP and RP are drawn parallel to BC, CA and AB.

To prove: $\text{BC}=\frac{1}{2}\text{QR}$

Proof: Since AR || BC and AB || RC [Given]

So, ABCR is a parallelogram. Therefore

AR = BC ...(i)

Also, AQ || BC and QB || AC

So, AQBC is a parallelogram.Therefore

QA = BC ...(ii)

Adding both side of (i) and (ii), we get

AR + QA = BC + BC

⇒ QR = 2BC

$\Rightarrow\text{BC}=\frac{\text{QR}}{2}$

$\therefore\text{BC}=\frac{1}{2}\text{QR}$

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