Question

ac is tangent to circle o at point a, and mab = 59. what is m∠acb? (not drawn to scale) a 121 b 21 c 31 d 29

Explanation:

Step1: Recall the tangent - radius property

The radius is perpendicular to the tangent at the point of tangency. So, $\angle OAC = 90^{\circ}$.

Step2: Use the central - inscribed angle relationship

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc. The central angle corresponding to arc $\overset{\frown}{AB}$ is $\angle AOB$ and $m\overset{\frown}{AB}=m\angle AOB = 59^{\circ}$. In $\triangle OBC$, $OB = OC$ (radii of the same circle), but we can also use the angle - relationship in $\triangle AOC$. Since $\angle AOB$ is an exterior angle of $\triangle BOC$ (if we consider the circle - related angles). We know that in $\triangle AOC$, we want to find $\angle ACB$. We know that $\angle AOB$ is the central angle for arc $\overset{\frown}{AB}$ and we use the fact that the angle between a tangent and a chord is half the measure of the intercepted arc.
The measure of the angle between a tangent and a chord is half the measure of the intercepted arc. So, $m\angle ACB=\frac{1}{2}m\overset{\frown}{AB}$.

Step3: Calculate the measure of $\angle ACB$

Since $m\overset{\frown}{AB} = 59^{\circ}$, then $m\angle ACB=\frac{59^{\circ}}{2}= 29.5^{\circ}\approx29^{\circ}$ (rounding to the nearest whole - number as per the multiple - choice options).

Answer:

D. 29