Chapter 10.7, Problem 12PSB

a.

To determine

Find the coordinates of the center of the circle.

Answer to Problem 12PSB

  $\left(-4.5,0.5\right)$

Explanation of Solution

Given information:

A circle is inscribed in a square with vertices $\left(-8,-3\right),\left(-1,-3\right),\left(-8,4\right)and\left(-1,4\right)$

Calculation:

Consider the given vertices, $\left(-8,-3\right),\left(-1,-3\right),\left(-8,4\right)and\left(-1,4\right)$ the figure will be,

  

  $\begin{array}{l}Thecentercoordinatesare,\\ y=\frac{y_2-y_1}{2},x=\frac{x_2-x_1}{2}\\ =\frac{-8+\left(-1\right)}{2},\frac{4+\left(-3\right)}{2}\\ -4.5,0.5\end{array}$

Hence, the coordinates of the center of the circle are $\left(-4.5,0.5\right)$ .

b.

To determine

Find the area of the circle.

Answer to Problem 12PSB

  $80squareunit$ .

Explanation of Solution

Given information:

A circle is inscribed in a square with vertices $\left(-8,-3\right),\left(-1,-3\right),\left(-8,4\right)and\left(-1,4\right)$

Calculation:

Consider the given vertices, $\left(-8,-3\right),\left(-1,-3\right),\left(-8,4\right)and\left(-1,4\right)$ the figure will be,

  

The diagonal length is the diameter of the circle,

  $\begin{array}{l}Diagonallength=\sqrt{{\left(-8-\left(-1\right)\right)}^2+{\left(4-\left(-3\right)\right)}^2}\\ =\sqrt{49+49}\\ =7\sqrt{2}\\ radius=\frac{7\sqrt{2}}{2}\end{array}$

  $\begin{array}{l}Area=\pi r^2\\ Area=\pi {\left(\frac{7\sqrt{2}}{2}\right)}^2\cong 80\\ Area=80squareunit\end{array}$

Hence, the area of the circle is $80squareunit$ .

c.

To determine

Find the radius of a circle circumscribed about the square.

Answer to Problem 12PSB

  $Radius=\frac{7}{2}$ .

Explanation of Solution

Given information:

A circle is inscribed in a square with vertices $\left(-8,-3\right),\left(-1,-3\right),\left(-8,4\right)and\left(-1,4\right)$

Calculation:

Consider the given vertices, $\left(-8,-3\right),\left(-1,-3\right),\left(-8,4\right)and\left(-1,4\right)$ the figure will be,

  

The circle radius is half of the length of the square side,

  $Radius=\frac{7}{2}$

Hence, the radius of a circle circumscribed about the square. $Radius=\frac{7}{2}$