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 the breadth. Q and R lie on the hypotenuse and P, S lie on the two different smaller sides of the triangle. What is the ratio of the areas of the rectangle and that of the triangle :

• A. $\sqrt{2}:1$
• B. $1:\sqrt{2}$
• C. $1:2$
• D. $\sqrt{3}:2$

Answer 1

The correct option is C $1:2$

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
$\therefore$ Area of square $PTUS=\frac{a^2}{2}$
$\therefore \frac{AreaofPTUS}{AreaofABCD}=\frac{\frac{a^2}{2}}{a^2}=\frac{1}{2}$
$\therefore \frac{AreasquarePQRS}{Area\Delta ABC}=\frac{1}{2}$