A circle touches all four sides of a quadrilateral ABCD. If $AB = 18 ext{ cm}$, $BC = 21 ext{ cm}$ and $AD = 15 ext{ cm}$, then find the length of $CD$.

Question

A circle touches all four sides of a quadrilateral ABCD. If $AB = 18 	ext{ cm}$, $BC = 21 	ext{ cm}$ and $AD = 15 	ext{ cm}$, then find the length of $CD$.

Accepted Answer

**Solution:**

1. Given that a circle touches all four sides of quadrilateral ABCD, it means ABCD is a tangential quadrilateral.

2. A key property of a tangential quadrilateral is that the sum of the lengths of opposite sides are equal:
   
   $$AB + CD = BC + AD$$

3. Substitute the known values:
   
   $$18 + CD = 21 + 15$$

4. Simplify the right side:
   
   $$18 + CD = 36$$

5. Solve for $CD$:
   
   $$CD = 36 - 18 = 18 	ext{ cm}$$

**Answer:** $18 	ext{ cm}$

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