Chapter 9.6, Problem 22WE

To determine

To calculate: The measure of third arc from the help of given information.

Answer to Problem 22WE

The measurement of third angle is $150°$ .

Explanation of Solution

Given information:

A secant and a tangent to a circle intersect in a 42° angle.

The two arcs of the circle intercepted by the secant and the tangent have measures in a ratio of 7:3.

O center is the center of the circle, where AC is a secant intersecting the circle at point B and AT is a tangent to the circle.

In the diagram we can see that,

  

As measure of arc CT and arc BT are in ratio 7:3 and also $\angle CAT=42°$ .

Assume that measure of arc CT is $7x$ and measure of arc BT is $3x$ .

As it is known that $\angle CAT=\frac{7x-3x}{2}$

Put the values

  $\begin{array}{l}\angle CAT=\frac{7x-3x}{2}\hfill \\ 42=\frac{7x-3x}{2}\hfill \end{array}$

Solving the above equation:

  $\begin{array}{l}42=\frac{7x-3x}{2}\hfill \\ 84=4x\hfill \\ 21=x\hfill \end{array}$

Thus measure of arc CT is

  $\begin{array}{l}=7x\hfill \\ =7\times 21\hfill \\ =147°\hfill \end{array}$

Thus measure of arc BT is

  $\begin{array}{l}=3x\hfill \\ =3\times 21\hfill \\ =63°\hfill \end{array}$

As it is known that $CT+BT+CB=360°$

So put the value

  $\begin{array}{l}CT+BT+CB=360°\hfill \\ 147+63+CB=360°\hfill \end{array}$

Solve the equation to get

  $\begin{array}{l}147+63+CB=360°\hfill \\ 210+CB=360°\hfill \\ CB=150°\hfill \end{array}$

Hence, the measurement of third angle is $150°$ .