CAT 2025 Slot 3 QA Question 9: ABCD is a trapezium in which $AB$ is parallel to $DC$, $AD$ is perpendicular to $AB$, and $AB = 3DC$. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area,

Question

ABCD is a trapezium in which $AB$ is parallel to $DC$, $AD$ is perpendicular to $AB$, and $AB = 3DC$. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area,

Accepted Answer

Since the circle touches both parallel sides, the height of the trapezium is twice the radius.
$	ext{Height} = 2 	imes 3 = 6$
Let $DC = x$. Then $AB = 3x$ and $AD = 6$. In a tangential quadrilateral, the sums of opposite sides are equal.
$AB + DC = AD + BC$
$3x + x = 6 + \sqrt{(3x - x)^2 + 6^2}$
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ABCD is a trapezium in which ABABAB is parallel to DCDCDC, ADADAD is perpendicular to ABABAB, and AB=3DCAB = 3DCAB=3DC. If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area, in sq. cm, of the trapezium is

ABCD is a trapezium in which $AB$ is parallel to $DC$ , $AD$ is perpendicular to $AB$ , and $AB = 3DC$ . If a circle inscribed in the trapezium touching all the sides has a radius of 3 cm, then the area, in sq. cm, of the trapezium is