Question

In the given figure O is the centre of the circle. ABCD is a quadrilateral where sides AB, BC, CD and DA touch the circle at E, F, G and H respectively. If AB = 15 cm, BC = 18 cm and AD = 24 cm, find the length of CD.

Answer 1

Given: O is the centre of the circle. ABCD is a tangential quadrilateral with sides AB, BC, CD, DA touching the circle at E, F, G, H respectively. AB = 15 cm, BC = 18 cm, AD = 24 cm. Find CD.

Step-wise calculation:

1. Let AE = AH = p   ...(Tangent segments from each vertex are equal)

BE = BF = q

CF = CG = r

DG = DH = s 

2. Then AB = AE + BE

= p + q

= 15

BC = BF + CF

= q + r

= 18

AD = AH + HD

= p + s

= 24 

And CD = CG + GD

= r + s

3. Add AB and CD:

AB + CD = (p + q) + (r + s)

= (p + s) + (q + r)

= AD + BC

This gives the standard tangential-quadrilateral relation AB + CD = AD + BC.

4. Solve for CD:

CD = AD + BC – AB

= 24 + 18 – 15

= 27