Question

an interior designer is hanging a circular clock for a client, as shown. the hanger at point b connects to the clock by two wires that are tangent to the clock at points a and c. if the radius of the clock is 20 cm and the distance from the top of the clock at point d to the hanger at point b is 5 cm, what is the length from point a to point b?

Explanation:

Step1: Identify right - triangle

Connect the center of the circle (assume it is the center of the clock) to point A. Since the wire is tangent to the circle at point A, the radius is perpendicular to the tangent at the point of tangency. Let the radius of the circle $r = 20$ cm and $BD=5$ cm. The distance from the center of the circle to point B is $r + 5$ cm.

Step2: Apply Pythagorean theorem

In right - triangle $ABE$ (where $E$ is the center of the circle), if the radius $EA=r = 20$ cm and $EB=r + 5=20 + 5=25$ cm. According to the Pythagorean theorem $AB=\sqrt{EB^{2}-EA^{2}}$.
\[

$$\begin{align*} AB&=\sqrt{(25)^{2}-(20)^{2}}\\ &=\sqrt{(25 + 20)(25 - 20)}\\ &=\sqrt{45\times5}\\ &=\sqrt{225}\\ & = 15 \end{align*}$$

\]

Answer:

15 cm