Question

in the diagram, \\( \overline { a c } \\) is a diameter of circle \\( m \\), and \\( m \angle c a b = 27 ^ { \circ } \\).

if a point within the circle is chosen at random, what is the probability that it lies within the shaded region?

write your answer as a fraction or decimal. do not round.

Explanation:

Step1: Use the property of the central angle and inscribed angle

The inscribed angle theorem states that an inscribed angle is half of the central angle subtended by the same arc. Here, \(\angle CAB\) is an inscribed angle and \(\angle CMB\) is the central angle subtended by arc \(CB\). So, \(m\angle CMB = 2\times m\angle CAB\).
Since \(m\angle CAB=27^{\circ}\), then \(m\angle CMB = 2\times27^{\circ}=54^{\circ}\).

Step2: Calculate the probability

The probability \(P\) that a randomly - chosen point in the circle lies in the shaded sector is given by the formula \(P=\frac{\text{measure of the central angle of the sector}}{\text{total measure of the central angle of the circle}}\).
The total measure of the central angle of a circle is \(360^{\circ}\).
So, \(P = \frac{54}{360}\).
Simplify the fraction: \(\frac{54\div18}{360\div18}=\frac{3}{20}=0.15\).

Answer:

\(0.15\) or \(\frac{3}{20}\)