Quantitative Aptitude Ques 1427

Question: In the given diagram, $ \Delta ABC $ is an isosceles right angled triangle, in which a rectangle is inscribed in such a way that the length of the rectangle is twice of breadth. Q and R lie on the hypotenuse; P and S lie on the two different smaller sides of the triangle. What is the ratio of the areas of the rectangle and that of triangle?

Options:

A) $ \sqrt{2}:1 $

B) $ 1:\sqrt{2} $

C) $ 1:2 $

D) $ \sqrt{3}:2 $

Show Answer

Answer:

Correct Answer: C

Solution:

  • PTUS is a square inscribed by a square ABCD. Let each side of the square ABCD be a. Then, area of square ABCD $ =a^{2} $ Also, $ PU=ST=a $ $ \frac{Areaof\square PTUS}{Areaof\square ABCD}=\frac{a^{2}/2}{a^{2}}=\frac{1}{2} $

$ \therefore $ $ \frac{AreaofPQRS}{2\times Areaof\Delta ABC}=\frac{1}{2} $ Now, $ ar\square PTUS=ar\Delta ABC $

$ \Rightarrow $ $ 2arPQRS=ar,\Delta ABC $

$ \therefore $ $ \frac{ar(PQRS)}{ar(\Delta ABC)}=\frac{1}{2} $

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