Question

Let ABCD be a quadrilateral in which AB is parallel to CD and perpendicular to AD , AB = 3CD, and the area of the quadrilateral is 4. If a circle can be drawan touching all the sides of quadrilateral , find its radius.

Answer 1

Explanation:

Let P,Q,R,S be the points of contact of in-circle with the sides AB, BC, CD, DA respectively. Since AD is perpendicular to AB and AB is parallel to DC, we see that AP = AS = SD = DR = r, the radius of the inscribed circle. 

Let BP = BQ = y and CQ = CR = x. 

Using AB = 3CD, we get r + y = 3(r + x)

Since the area of ABCD is 4. we also get

4 = 1/2AD(AB CD) = 1/2(2r)(4(r + x)).

Thus we obtain r (r + x) = 1.

Using Pythagoras theorem, we obtain BC² = BK² + CK² . 

However BC = y + x ,BK = y - x and CK = 2r. 

Substituting these and simplifying, we get xy = r² . 

But r + y = 3 (r + x) gives y = 2r + 3x. 

Thus r² = x(2r + 3x) and this simplifies to get (r – 3x)(r + x) = 0. 

We conclude that r = 3x. 

Now the relation r(r + x) = 1 implies that 4r² = 3, gives r = √3/2.