Chapter 7.1, Problem 69E

To determine

Find $m\angle A$ .

Answer to Problem 69E

  $\left(C\right)40°$

Explanation of Solution

Given information:

Triangle $ABC$ is inscribed in the circle $O$ , and $\overline{CD}$ is tangent to circle $O$ at point. If $m\angle BCD=40°$

  $\left(A\right)60°\left(B\right)50°\left(C\right)40°\left(D\right)30°\left(E\right)20°$

Calculation:

Here, we will consider:

Now, draw a line perpendicular to $CD$ at $C$ , passing through $O$ and let it meet circum circle at $E$ :

As,

  $\begin{array}{l}CD\perp CD,wehavem\angle ECD=90°\\ m\angle ECB=m\angle ECD-m\angle BCD\\ m\angle ECB=90°-40°\\ m\angle ECB=50°\\ \because m\angle CBE=90°\\ \therefore m\angle CEB=90°-50°\\ =40°\end{array}$

As, angles on the same segment of circle subtended by a chord are same.

  $\begin{array}{l}m\angle CAB=m\angle CEB\\ m\angle CAB=40°\\ \therefore m\angle A=40°\end{array}$

Hence, the correct answer is $option\left(C\right)$ .