Chapter 10.6, Problem 9PSA

To determine

Find the value for the given angle using diagram.

Answer to Problem 9PSA

The value of the angle $m\angle P={42}^{°}$ ..

Explanation of Solution

Given:

The given diagram is

  

The given angle is $m\angle P$ .

Calculation:

In diagram, draw a line $\overline{CA}$ and $\overline{CB}$ . $C$ is the center of the circle.

  

A tangent of a circle is perpendicular to the radius of the circle.

  $\angle PAC=\angle PBC={90}^{°}$ .

  $PACB$ is a quadrilateral.

  $\angle P+\angle A+\angle C+\angle B={360}^{°}$

  $\angle P=6x^{°},\angle C={\left(15x+33\right)}^{°}$ and $\angle A=\angle B={90}^{°}$ .

  $\begin{array}{l}6x^{°}+{90}^{°}+{\left(15x+33\right)}^{°}+{90}^{°}={360}^{°}\\ \Rightarrow 21x+{213}^{°}={360}^{°}\\ \Rightarrow 21x+{213}^{°}-{213}^{°}={360}^{°}-{213}^{°}\\ \Rightarrow 21x={147}^{°}\\ \Rightarrow \frac{21}{21}x=\frac{{147}^{°}}{21}\\ \Rightarrow x=7^{°}\end{array}$

Now,

  $m\angle P=6x=6\cdot 7^{°}={42}^{°}$

Hence the value of the angle $m\angle P={42}^{°}$ .