Chapter 10.6, Problem 13PSB

To determine

Find the radius of the circle using given statement.

Answer to Problem 13PSB

The radius of the circle is $15$ unit.

Explanation of Solution

Given:

The given statement is that the rectangle inscribed in a circle with dimension $18$ by $24$ .

Calculation:

Draw a diagram using the given statement.

  

In diagram, $C$ is the center of the circle and the rectangle.

  $\begin{array}{l}\overline{PQ}=\overline{SR}=24\\ \overline{PS}=\overline{QR}=18\end{array}$

Use theory, the angle inscribed in a semi-circle is a right angle.

  $\angle PSR={90}^{°}$ .

Now triangle $\Delta PSR$ is a right angle triangle.

Use Pythagorean formula.

  $\text{hypotenuse}=\sqrt{{\text{perpendicular}}^2+{\text{base}}^2}$ .

  $\begin{array}{l}PR=\sqrt{P{S}^2+S{R}^2}\\ =\sqrt{{18}^2+{24}^2}\\ =\sqrt{324+576}\\ =\sqrt{900}\\ =30\end{array}$

  $PR$ is the diagonal and diameter of the rectangle and circle respectively.

  $PC$ is the radius of the circle.

  $PC=\frac{PR}{2}=\frac{30}{2}=15$

Hence the radius of the circle is $15$ unit.