Question

question 4 of 10
in the diagram below, $overline{ab}$ and $overline{bc}$ are tangent to $odot o$. what is the measure of $widehat{adc}$?
a. $200^{circ}$
b. $220^{circ}$
c. $160^{circ}$
d. $180^{circ}$

Explanation:

Step1: Recall tangent - radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So $\angle OAB=\angle OCB = 90^{\circ}$.

Step2: Find $\angle AOC$ in quadrilateral $OABC$

In quadrilateral $OABC$, the sum of interior angles is $360^{\circ}$. Given $\angle ABC=40^{\circ}$, $\angle OAB = 90^{\circ}$, $\angle OCB=90^{\circ}$. Let $\angle AOC=x$. Then $90^{\circ}+90^{\circ}+40^{\circ}+x = 360^{\circ}$, so $x=140^{\circ}$.

Step3: Find the measure of arc $ADC$

The measure of an arc is related to the central - angle that subtends it. The measure of the major arc $ADC$ and the central - angle $\angle AOC$ (the minor - arc central - angle) are related by the fact that the sum of the measures of a major arc and its corresponding minor arc is $360^{\circ}$. If the measure of the minor arc $\overset{\frown}{AC}$ is $140^{\circ}$, then the measure of the major arc $\overset{\frown}{ADC}=360^{\circ}- 140^{\circ}=200^{\circ}$.

Answer:

A. 200°