Chapter 9.1, Problem 15WE

To determine

To calculate: Find the length of $\overline{AB}$.

Answer to Problem 15WE

  $20.78\text{ cm}$ .

Explanation of Solution

Given information: For each exercise draw $\odot O$ with radius 12 cm. Then draw radii $\overline{OA}$ and $\overline{OB}$ to form an angle with the measure named. $m\angle AOB=120°$ .

Calculation:

  

Since, from the figure, $AO=OB=12\text{ cm}$ , then $\angle A=\angle B$ .

Now, $\angle A+\angle B+\angle O=180°$ .

  $\Rightarrow 2\angle A+120°=180°,$

  $\Rightarrow \angle A=30°$ , and $\angle B=30°$ .

Therefore, the triangle is an isosceles tringle.

Now, from the sine rule, we can write:

  $\frac{AO}{\sin B}=\frac{AB}{\sin O}$ ,

  $\Rightarrow \frac{12\text{ cm}}{\sin 30°}=\frac{AB}{\sin 120°},$

  $\Rightarrow AB=20.78\text{ cm}$ .