Question

For each exercise draw $\odot O$ with radius $12$. Then draw radii $\overline{OA}$ and to $\overline{OB}$ form an angle with the measure named. Find the length of $\overline{AB}$.

d. $m\angle AOB=120°$

Short answer

$AB=20.78$.

Step-by-step solution

Step 1. Given information.  

Step 2. Concept used.

Isosceles triangle concept. We also use sine formula.

Step 3. Find value.

Since from the figure,

$\begin{array}{c}AO=OB\\ AO=12\\ OB=12\\ \angle A=\angle B\\ \end{array}$

In role="math" localid="1648892994173" $\Delta AOB$, a isosceles triangle.

$\begin{array}{c}\angle A+\angle B+\angle O={180}^{\circ }\\ 2\angle A+{120}^{\circ }={180}^{\circ }\\ \angle A={30}^{\circ }\\ \end{array}$

So, $\angle B=30°$

Now, by sine formula:

$\begin{array}{c}\frac{AO}{\sin B}=\frac{AB}{\sin O}\\ \frac{12}{\sin {30}^{\circ }}=\frac{AB}{\sin {120}^{\circ }}\\ AB=20.78\end{array}$

Therefore, this length of $AB=20.78$.