Chapter 12, Problem 8CT

To determine

Find the angle between the legs of the compass.

Answer to Problem 8CT

  $112.88°$

Explanation of Solution

Given:

A compass with legs 3 in. long is opened to measure the diameter of a circle. The diameter is 5 in.

Calculation:

Since the legs of the compass are equal , the triangle with base as the diameter of the circle is isosceles triangle.

Hence , the perpendicular from the vertex of the compass bisects the diameter.

Represent the situation and solve :

  

Find the angle ABD:

  $\begin{array}{l}\cos \angle ABD=\frac{\text{base}}{hypotenuse}\\ \Rightarrow \cos \angle ABD=\frac{2.5}{3}\\ \Rightarrow \angle ABD={\cos }^{-1}(0.8333)\\ \Rightarrow m\angle ABD=33.56°\end{array}$

Now,

  $\begin{array}{l}m\angle ABD+m\angle BAD+m\angle ADB=180°\\ \Rightarrow m\angle BAD=180°-90°-33.56°\\ \Rightarrow m\angle BAD=56.44°\\ \Rightarrow m\angle BAC=2m\angle BAD=112.88°\end{array}$

So, the angle between the legs of compass is $112.88°$ .