Chapter 9.5, Problem 21WE

To determine

To find: The value of x and angle A .

Answer to Problem 21WE

The value of x and angle A are 10 and 55 respectively.

Explanation of Solution

Given:

  $m\angle D=75$

  $mAB=x^2$

  $mBC=5x$

  $mCD=6x$

Calculation:

Inscribed angle in circle is half of its intercepted arc.

Thus,

  $\begin{array}{l}m\angle D=\frac{1}{2}\left(mAB+mBC\right)\\ 75=\frac{1}{2}\left(x^2+5x\right)\\ x^2+5x-150=0\\ \left(x-10\right)\left(x+15\right)=0\\ x=10,-15\end{array}$

The value of x cannot be negative

Thus,

  $x=10$

The angle A is calculated as follows:

  $\begin{array}{c}m\angle A=\frac{1}{2}\left(mCD+mBC\right)\\ =\frac{1}{2}\left(6x+5x\right)\\ =\frac{1}{2}\left(6\left(10\right)+5\left(10\right)\right)\\ =55\end{array}$

Thus, the value of x and angle A are 10 and 55 respectively.