Question

question 10 of 10
in the diagram below, \\( \overline { a b } \\) and \\( \overline { b c } \\) are tangent to \\( \odot o \\). what is the measure of \\( \widehat { a c } \\) ?
a. \\( 180 ^ { \circ } \\)
b. \\( 90 ^ { \circ } \\)
c. \\( 150 ^ { \circ } \\)
d. \\( 60 ^ { \circ } \\)

Explanation:

Step1: Recall the property of tangents to a circle

If \(AB\) and \(BC\) are tangent to \(\odot O\), then \(OA\perp AB\) and \(OC\perp BC\), so \(\angle OAB = 90^{\circ}\) and \(\angle OCB=90^{\circ}\).

Step2: Use the sum of interior angles of a quadrilateral

The sum of interior angles of quadrilateral \(OABC\) is \(360^{\circ}\). Let \(\angle AOC\) be the central angle corresponding to arc \(\widehat{AC}\). We know \(\angle B = 30^{\circ}\), \(\angle OAB = 90^{\circ}\), \(\angle OCB = 90^{\circ}\).
By the formula \(\angle OAB+\angle B+\angle OCB+\angle AOC=360^{\circ}\), substituting the values: \(90^{\circ}+30^{\circ}+90^{\circ}+\angle AOC = 360^{\circ}\).
Simplify the left - hand side: \(210^{\circ}+\angle AOC=360^{\circ}\).
Solve for \(\angle AOC\): \(\angle AOC=360^{\circ}-(90^{\circ} + 30^{\circ}+90^{\circ})=150^{\circ}\).
Since the measure of an arc is equal to the measure of its central angle, the measure of \(\widehat{AC}\) is equal to the measure of \(\angle AOC\).

Answer:

C. \(150^{\circ}\)