Chapter 9.5, Problem 13PSB

Solution

a

To find: Area of sector TRV to the nearest tenth

Area of sector TRV = $42.4$ square units.

Given information:

Here, we have given that measure of $\stackrel{⌢}{VW}$ is 120 and RW is 9.

Formula used: Area of sector = $\frac{\theta }{360°}\pi r^2$

Calculation: The measure o0f minor arc is equal to the measure of its central angle.

  $\therefore$ m $\stackrel{⌢}{VW}$ = $\angle VRW$

  $\Rightarrow \angle VRW=120°$

Now, $\angle VRW+\angle VRT=180°$ (Linear Pair Angles)

  $\begin{array}{l}\Rightarrow 120°+\angle VRT=180°\\ \Rightarrow \angle VRT=180°-120°\\ \Rightarrow \angle VRT=60°\end{array}$

As, RT = RW (Radius of the circle)

  $\Rightarrow$ RT = $9$

So, Area of sector TRV = $\frac{\theta }{360°}\pi r^2$

  $\begin{array}{l}\Rightarrow \frac{60°}{360°}\times \frac{22}{7}\times {(}^9\\ \Rightarrow \frac{1}{6}\times \frac{22}{7}\times 81\\ \Rightarrow \frac{1782}{42}\\ \Rightarrow 42.42\end{array}$

Nearest tenth = $42.4$ square units

b

To find: Difference between length of $\overline{TW}$ and length of $\stackrel{⌢}{VW}$ to the nearest tenth.

The difference between the segments is 111 units.

Given information:Here, we have given that measure of $\stackrel{⌢}{VW}$ is 120 and RW is 9.

Formula used: $\overline{TW}=\overline{TR}+\overline{RW}$

Calculation: $\overline{TR}=\overline{RW}$ (Radius of circle)

  $\Rightarrow \overline{TR}=9$

So, $\overline{TW}=9+9=18$

Now, difference = $\stackrel{⌢}{VW}-\overline{TW}=120-9=111$

Nearest tenth = 111.0