question_answer

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?

A) \[\sqrt{2}:1\]                 

B) \[1:\sqrt{2}\]

C) \[1:2\]   

D) \[\sqrt{3}:2\]

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,     \[PU=ST=a\]

\[\frac{\text{Area}\,\,\text{of}\,\,\square \,\,PTUS}{\text{Area}\,\,\text{of}\square \,\,ABCD}=\frac{{{a}^{2}}/2}{{{a}^{2}}}=\frac{1}{2}\]

\[\therefore \]      \[\frac{\text{Area}\,\,\text{of}\,\,\,\,PQRS}{2\times \text{Area}\,\,\text{of}\,\,\Delta ABC}=\frac{1}{2}\]

Now,     \[ar\,\,\square \,\,PTUS=ar\,\,\Delta ABC\]

\[\Rightarrow \]   \[2ar\,\,PQRS=ar\,\Delta ABC\]

\[\therefore \]      \[\frac{ar\,\,(\,\,PQRS)}{ar\,\,(\Delta ABC)}=\frac{1}{2}\]