IIT-JEE 2007 · Paper 2 · Mathematics Q51

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MathematicsSingle correct+3 / -1Question 51 of 66

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

\frac32

Answer and worked solution

Official answer: B

Set up. ABCD is a right trapezoid with $CD=a$ , $AB=2a$ and height $AD$ . A circle touching all four sides has diameter equal to the distance between the parallel sides: $AD=2r$ .

Tangential quadrilateral. $AB+CD=AD+BC$ gives $BC=3a-2r$ . Also $BC^2=AD^2+(AB-CD)^2=4r^2+a^2$ :
$(3a-2r)^2=4r^2+a^2\implies8a^2=12ar\implies a=\frac{3r}2.$

Area. $\frac12(a+2a)(2r)=3ar=\frac92r^2=18$ , so $r=2$ .

(B)

Topics:Tangents and normals Distance and section formulae

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IIT-JEE 2007 · Paper 2, Mathematics question 51