Question

In the figure, AB is a diameter and AP is a tangent to the circle. If PB = 9 cm and PQ = 4 cm, find the length of AP. What is the radius of the circle?

Answer 1

From the figure, $PQ\times PB=P{A}^2$
$\Rightarrow P{A}^2=4\times 9=36\Rightarrow PA=\sqrt{36}=6cm$
In right triangle APB, $A{P}^2+A{B}^2=P{B}^2$
$\Rightarrow 6^2+A{B}^2=9^2$
$\Rightarrow A{B}^2=9^2-6^2=81-36=45\Rightarrow AB=\sqrt{45}=\sqrt{9\times 5}=3\sqrt{5}$
Thus, radius of the circle $=\frac{AB}{2}=\frac{3\sqrt{5}}{2}cm$