Chapter 9.5, Problem 8WE

To determine

To find: The values of x, y and z .

Answer to Problem 8WE

  $x=80°$, $y=40°$ and $z=60°$

Explanation of Solution

Given:

The figure:

  

Concept used:

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

Calculation:

Let us name the quadrilateral inscribed in the circle as PQRS.

  

As the three sides are equal, ( $PQ=QR=RS$ ) the angle intercepted by the arcs are also equal.

  ( $\stackrel{⌢}{PQ}=\stackrel{⌢}{QR}=\stackrel{⌢}{RS}$ ) and $\stackrel{⌢}{PQ}=80°$

As $x=m\stackrel{⌢}{SR}$

Therefore, $x=80°$.

Now, the measure of an inscribed angle is equal to half the measure of its intercepted arc.

Therefore, $y=\frac{1}{2}\left(\stackrel{⌢}{QR}\right)=\frac{1}{2}\left(80°\right)=40°$

Also, as there are three arc of same angle, $\stackrel{⌢}{PQ}=\stackrel{⌢}{QR}=\stackrel{⌢}{RS}$ so,

  $\begin{array}{l}\stackrel{⌢}{PS}=360°-(\stackrel{⌢}{PQ}+\stackrel{⌢}{QR}+\stackrel{⌢}{RS})\\ \stackrel{⌢}{PS}=360°-(80°+80°+80°)\\ \stackrel{⌢}{PS}=360°-(240°)\\ \stackrel{⌢}{PS}=120°\end{array}$

And the measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

Therefore, $z=\frac{1}{2}\left(\stackrel{⌢}{PS}\right)=\frac{1}{2}\left(120°\right)=60°$.

Conclusion:

  $x=80°$, $y=40°$ and $z=60°$ .