Chapter E, Problem 9.12EX

To determine

To find: the angle which is equal to $m\angle AXZ$ .

Answer to Problem 9.12EX

The angle which is equal to $m\angle AXZ$ is $\frac{1}{2}\angle XOZ$ .

Explanation of Solution

Given information:

  $XA$ is tangent to O at X.

  

Calculation:

Let $\angle XOY=x$   ... (1)

In $\Delta XOM$ , apply property of triangle to calculate measure of $\angle OXM$ .

  $\begin{array}{l}\angle XOM+\angle OMX+\angle OXM=180\\ \angle XOY=\angle XOM=x\\ x+90+\angle OXM=180\\ \angle OXM=90-x\end{array}$

Here, XA is tangent to the circle.

  $\begin{array}{l}\angle OXA=90°\\ \angle OXM+\angle MXA=90°\\ 90°-x+\angle MXA=90°\\ \angle MXA=x\end{array}$   ... (2)

By equation (1) and (2):

  $\angle XOY=\angle MXA$

From figure, it is clear that:

  $\angle MXA=\angle AXZ$

Hence

  $\angle XOY=\angle AXZ$   ... (3)

Now

  $\begin{array}{l}\angle XOZ=2\angle XOY\\ \angle XOY=\frac{1}{2}\angle XOZ\end{array}$   ... (4)

By equation (3) and (4):

  $\angle AXZ=\frac{1}{2}\angle XOZ$