Let `A B C D` be a quadrilateral with are `18` , side `A B` parallel to the side `C D ,a n dA B=2C D` . Let `A D` be perpendicular to `A Ba n dC D` . If a circle is drawn inside the quadrilateral `A B C D` touching all the sides, then its radius is `3` (b) 2 (c) `3/2` (d) 1

Options

A. 3
B. 2
C. `(3)/(2)`
D. 1

Answer

B

Explanation

`18=(1)/(2) (3a)(2r)`
`rArr "ar" = 6`
Line `y=-(2r)/(alpha)(x-2alpha)` is tangent to circle
`(x-r)^(2)+(y-r)^(2)=r^(2)`
`2alpha=3r,ar=6 and r=2`

Alternative Solution
`(1)/(2)(x+2x)xx2r=18`
`xr=6" "...(i)`
In `Delta AOB, tantheta = (x-r)/(r)`
and "in" `Delta DOC`,
`tan(90^(@)-theta)=(2x-r)/(r)`
`therefore (x-r)/(r)=(r)/(2x-r)`
`rArrx(2x-3x)=0`
`rArr x=(3r)/(2)" "...(ii)`
From Eqs. (i) and (ii), we get
r = 2