Question

complete the statements below. give your answers as exact values, not decimal approximations. (a) find the ratio of the area of the sector to the area of the circle. \\(\frac{\text{area of sector}}{\text{area of circle}} = \square\\) (b) from the figure we see that \\(m \angle acb = 45^\circ\\). the measure of \\(\angle acb\\) is the measure of \\(\overarc{ab}\\). so, \\(m \overarc{ab} = \square^\circ\\).

Explanation:

Step1: Solve part (a)

The ratio of the area of a sector to the area of the circle is equal to the ratio of the central angle of the sector to the total angle of the circle (which is \(360^\circ\)). The central angle of the sector \(\angle ACB\) is \(45^\circ\). So the ratio \(\frac{\text{Area of sector}}{\text{Area of circle}}=\frac{45^\circ}{360^\circ}\). Simplify this fraction: \(\frac{45}{360}=\frac{1}{8}\).

Step2: Solve part (b)

The measure of a central angle is equal to the measure of its intercepted arc. Here, \(\angle ACB\) is a central angle intercepting arc \(AB\), so the measure of arc \(AB\) (denoted as \(m\overset{\frown}{AB}\)) is equal to the measure of \(\angle ACB\), which is \(45^\circ\).

Answer:

(a) \(\frac{1}{8}\)
(b) \(45\)