Question

Let $ABCD$ be a quadrilateral with area $18$, with side $AB$ parallel to the side $CD$ and $AB=2CD$. Let $AD$ be perpendicular to $AB$ and $CD$. If a circle is drawn inside the quadrilateral $ABCD$ touching all the sides, then its radius is:

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

Answer 1

Answer: (B)

$2$

Given $AB=2CD$
Let , $CD=a$ and $AD=b$
Area of Trapezium = $\frac{1}{2}\left(\text{Sum of Lengths of parallel sides}\right)\text{(Distrance between Parallel Sides)}$

So, area of trapezium $ABCD=\frac{1}{2}\times (a+2a)\times b$$=\frac{3}{2}ab$
So, $\frac{3}{2}ab=18$
$ab=12$ ----(1)
$r=\frac{b}{2}=\frac{29}{3}$ ------(2)
from (1) & (2),
$a=3$ and $b=4$
So, radius $=\frac{b}{2}=2$