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