Chapter 9, Problem 20RP

a.

To determine

To find: The measure of arc $\stackrel{⌢}{RTC}$ .

Answer to Problem 20RP

The measure of arc $\stackrel{⌢}{RTC}$ is $180°$ .

Explanation of Solution

Given information:

RECT is a rectangle,

Side $RE=6,$

Side $EC=8.$

Formula used:

  $m\stackrel{⌢}{RTC}=2m\angle REC$

Calculation:

  

The measure of arc RTC is two times the measure of arc REC.

  $m\stackrel{⌢}{RTC}=2m\angle REC=2({90}^{\circ })={180}^{\circ }$

b.

To determine

To calculate: The length of arc $\stackrel{⌢}{RTC}$ .

Answer to Problem 20RP

The length of arc $\stackrel{⌢}{RTC}$ is $5\pi$ .

Explanation of Solution

Given information:

RECT is a rectangle,

Side $RE=6,$

Side $EC=8.$

Formula used:

  $l(\stackrel{⌢}{RTC})=\frac{{180}^{\circ }}{{360}^{\circ }}(2\pi r)$

Calculation:

  

  $l(\stackrel{⌢}{RTC})=\frac{{180}^{\circ }}{{360}^{\circ }}(2\pi \ast 5)=5\pi$

c.

To determine

To calculate: The area of shaded region.

Answer to Problem 20RP

The area of shaded region is $30.5$ .

Explanation of Solution

Given information:

RECT is a rectangle,

Side $RE=6,$

Side $EC=8.$

Formula used:The below theorem is used:

Pythagoras theorem states that “In a right angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides”.

  

In right angle triangle,

  $a^2+b^2\text{= }{c}^2$

Area of Circle: $A=\pi r^2$

r = radius of circle.

Area of rectangle: $A=length*width$

Calculation:

  

The Pythagoras theorem is applied to calculate the radius of a circle.

In right angle triangle REC , we get

  ${\left(RC\right)}^2={\left(RE\right)}^2+{\left(EC\right)}^2$

  $\begin{array}{l}{\left(RC\right)}^2={\left(6\right)}^2+{\left(8\right)}^2\\ {\left(RC\right)}^2=36+64\\ {\left(RC\right)}^2=100\\ \therefore RC=\sqrt{100}=10\\ r=\frac{d}{2}=\frac{RC}{2}=\frac{10}{2}=5\end{array}$

  $A_C=\pi r^2=\pi {(5)}^2=25\pi \approx 78.5$

  $A_R=length*width=6*8=48$

Area of shaded region:

  $A_{ShadedRegion}=A_C-A_R=78.5-48\approx 30.5$