H4. The diagram shows a circle and a trapezium $A B C D$ in which $A D$ is parallel to $B C$ and $A B=D C$. All four sides of $A B C D$ are tangents of the circle. The circle has radius 4 and the area of $A B C D$ is 72 .
What is the length of $A B$ ?
Solution
Note that the height of the trapezium is equal to the diameter of the circle $=8$.
Since the area of the trapezium is 72 , we have $\frac{A D+B C}{2} \times 8=72$ and so $A D+B C=18$.
Let $E, F, G$ and $H$ be the points on $A B, B C, C D$ and $D A$ respectively where the circle is tangent to the sides of the trapezium, as shown in the diagram below.

Since the two tangents to a circle from any external point are equal in length, we know that $A E=A H$.
Similarly, $B E=B F, C F=C G$ and $D G=D H$.
Then $A B+D C=(A E+B E)+(C G+D G)=A H+B F+C F+D H=A D+B C$, which we know is equal to 18 .
Since $A B=D C$, we have $A B+D C=2 A B=18$ and hence $A B=9$.