Chapter 10.6, Problem 24PSC

To determine

Find the value of the given arc using diagram.

Answer to Problem 24PSC

The value of $mArc\left(HM\right)={120}^{°}$ .

Explanation of Solution

Given:

The given diagram is

  

$\text{RHOM}$ is a rhombus. Given arc is $mArc\left(HM\right)$ .

Calculation:

Let the $mArc\left(HM\right)=2x$

Inscribed angle $=\frac{1}{2}$ intercepted arcs

  $m\angle HOM=\frac{1}{2}Arc\left(HM\right)=\frac{1}{2}\cdot 2x=x$

Opposite angles of a rhombus are equal in measurement.

  $m\angle HRM=m\angle HOM=x$

The sum of the angle formed by the tangents and measurement of minor intercepted arc is ${180}^{°}$ .

  $\begin{array}{l}x+2x={180}^{°}\\ \Rightarrow 3x={180}^{°}\\ \Rightarrow \frac{3}{3}x=\frac{{180}^{°}}{3}\\ \Rightarrow x={60}^{°}\end{array}$

  $mArc\left(HM\right)=2x=2\cdot {60}^{°}={120}^{°}$

Hence the value of $mArc\left(HM\right)={120}^{°}$ .