Chapter 9.4, Problem 21WE

To determine

To Find: The measure of arc by using trigonometry.

Answer to Problem 21WE

The measure of arc is approx. $74°$ .

Explanation of Solution

Given information:

The circle contain chords $XY=12cm$ .

Radius of the circle is 10 cm.

Consider the figure as below:

  

In above figure circle contain chords $XY=12$ , point O is a center point of circle.

Radius of the circle OX= OY=10

A line segment OM is a perpendicular segment from O to chord XY and it bisects NA XOY into two equal parts which means NA XOZ= NA YOZ .

In circle M is the midpoint of the chord XY which means $XY=XM+YM$ and $XM=YM$ or it can be defined as

  $XM=\frac{XY}{2}$

If $XY=12$ then:

  $\begin{array}{l}\begin{array}{l}XM=\frac{XY}{2}\hfill \\ XM=\frac{12}{2}\hfill \end{array}\\ XM=6\end{array}$

To find the measure of angle $\angle XOZor\angle XOM$

Use trigonometric identity:

NA in form right triangle ΔXMO

Here perpendicular $XM=6$ , hypotenuse $OX=10$

  $\begin{array}{l}\sin \theta =\frac{6}{10}\\ \sin \theta =\frac{3}{5}\\ \theta ={\sin }^{-1}\left(\frac{3}{5}\right)\end{array}$

Here, $\angle XOZ={\sin }^{-1}\left(\frac{3}{5}\right)$ .

But angle $\angle XOY$ which has to be found is just double of $\angle XOZ={\sin }^{-1}\left(\frac{3}{5}\right)$

This means the angle is:

  $\begin{array}{l}\angle XOY=2\angle XOZ\hfill \\ \angle XOY=2\left({\sin }^{-1}\left(\frac{3}{5}\right)\right)\hfill \end{array}$

Here measure of ${\sin }^{-1}\left(\frac{3}{5}\right)=36.869897645844$

  $\begin{array}{l}\angle XOY=2\left({\sin }^{-1}\left(\frac{3}{5}\right)\right)\hfill \\ \angle XOY=2\times \left(36.869897645844\right)\hfill \\ \angle XOY=73.739795291688\hfill \\ \angle XOY\approx 74\hfill \end{array}$

If $\angle XOY\approx 74°$ then also $arcXZY\approx 74°$ .

According to theorem, In the same circle if two minor arc are congruent only when their central angles are congruent.

Therefore, measure of arc is approx. 74° .