CPM Homework Help : INT2 Problem 12-83

In a circle, chord $\overline{AB}$ has length $10$ units, while $m\overarc{AB} = 60°$ . What is the area of the circle? Draw a diagram and show all work.

Step 1:

The measure of the intercepted arc is equal to the central angle.
$\overarc{AB} = 60° = ∠C$

Step 2:

$AC=BC$ , because both are radii, so $ΔABC$ is isosceles.

Isosceles triangles have equal base angles, so $∠A = ∠B$ .

$∠A + ∠B + ∠C = 180°$
but $∠C = 60°$ , $∠A = ∠B$
so $∠B + ∠B + 60° = 180°$
$2∠B = 120°$
$∠B = 60° = ∠A$

Circle, center C, with points, A, &, B, line segments from, C, to, B, from, C, to, A, from, A, to, B. Segment, A, B, labeled, 10, arc, A, B, labeled, 60 degrees.

Answer:

$π(10)^2 = 100π = 314.16$ units

Step 3:

Because the angles are all equal, the triangle is equilateral.

So the radius ( $AC$ ) is $10$ .