## Area of Semicircle Found From Lines at Right Angles

The diagram shows a semicircle with two lines drawn at right angles. What is the area of the semicircle?

Complete the triangle ABC. From C draw a line to the end of the dimater at D. Triangle ACD is a right angled triangle.
If
$\angle BAC =x$
then
$\angle BCA = 90-x$
and
$\angle DCB=90-(90-x)=x, \: \angle BDC=90-x$
.
Triangles ABC and BDC have three identical angles, so are similar triangles.
Then
$\frac{AB}{BC}=\frac{BC}{BD} \rightarrow BD=\frac{BC \times BC}{AB}=\frac{16^2}{8}=32$