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MAT Exam 2022 Paper

Question : 114 of 200

Marks: +1, -0

\mathrm{AP}:\mathrm{BP}=1:4

\mathrm{BQ}:\mathrm{CQ}=2:3

Solution:

Concept:

When two triangles have the same included angle, their areas are proportional to the product of the ratios of the corresponding adjacent sides.

Explanation:

Let the area of

\triangle ABC

n

Given

AP:BP = 1:4

, so

BP = \frac{4}{5}AB

Given

BQ:CQ = 2:3

, so

BQ = \frac{2}{5}BC

\triangle BPQ

and

\triangle ABC

share the same angle at

B

Therefore,

\text{ar}(\triangle BPQ) = \frac{BP}{AB} \times \frac{BQ}{BC} \times \text{ar}(\triangle ABC)

So,

\text{ar}(\triangle BPQ) = \frac{4}{5} \times \frac{2}{5} \times n = \frac{8}{25}n

Area of quadrilateral

PQCA = \text{ar}(\triangle ABC) - \text{ar}(\triangle BPQ)

Thus,

\text{ar}(PQCA) = n - \frac{8}{25}n = \frac{17}{25}n

Hence,

\frac{\text{ar}(\triangle BPQ)}{\text{ar}(PQCA)} = \frac{\frac{8}{25}n}{\frac{17}{25}n} = \frac{8}{17}

So the required ratio is

8:17

Answer:

\frac{8}{17}

, i.e. option D.

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