Question

If triangle ABC is similar to triangle APQ and ar ( APQ) = 4 ar ( ABC)
then the ratio of BC to PQ is
(a) 2:1
(b) 1:2
(c) 1:4
(d) 4:1​

Answer 1

Given:

Δ ABC ~ Δ APQ

Ar(Δ APQ) = 4 Ar (Δ ABC)

To find:

The ratio of BC to PQ

Solution:

We have,

Ar(Δ APQ) = 4 Ar (Δ ABC)

⇒ $\frac{Ar (\triangle APQ)}{Ar (\triangle ABC)} = \frac{4}{1}$

⇒ $\frac{Ar (\triangle ABC)}{Ar (\triangle APQ)} = \frac{1}{4}$ .... (i)

Now we know that the ratio of the two similar triangles is equal to the ratio of the squares of their corresponding sides.

Since ΔABC ~ ΔAPQ, therefore we can write

$\frac{Ar (\triangle ABC)}{Ar (\triangle APQ)} = [\frac{BC}{PQ}]^2$

substituting from (i)

$\implies \frac{1}{4} = [\frac{BC}{PQ}]^2$

taking square roots on both sides

$\implies \sqrt{ \frac{1}{4}} = \sqrt{[\frac{BC}{PQ}]^2}$

$\implies \bold{\frac{BC}{PQ} = \frac{1}{2}}$ ← option (b)

Thus, the ratio of BC to PQ is 1:2.

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