Question

In the figure, $AB\perp BC,$ $BC\perp CD,$ and $BC$ is tangent to the circle with center $O$ and diameter $AD.$ In which one of the following cases is the area of $ABCD$ an integer?

$AB=3,$ $CD=1$

$AB=5,$ $CD=2$

$AB=7,$ $CD=3$

$AB=9,$ $CD=4$

$AB=11,$ $CD=5$

Answer: D

Concepts:circle tangency power of a point trapezoid area

Difficulty rating: 2280

Small Hint:

Let the tangent point be $M$ and use the diameter to form a rectangle inside the trapezoid

Big Hint:

Power of point $B$ relates half of $BC$ to $AB$ and $CD$

Solution:

The tangent point is the midpoint of $BC,$ and the tangent-secant relation gives $(\frac{BC}{2})^2=AB\cdot CD.$ Thus $BC=2\sqrt{AB\cdot CD},$ and the trapezoid area is $\begin{aligned} \text{Area} &=\frac{AB+CD}{2}BC\\ &=(AB+CD)\sqrt{AB\cdot CD}. \end{aligned}$ Only $AB=9,\ CD=4$ makes the product under the radical a square; the area is $13\cdot6=78.$

Thus the correct answer is D.

1988 AMC 12 Problem 27

Answer: D

Difficulty rating: 2280

Small Hint:

Big Hint:

Solution:

Problem 27 in Other Years