Chapter 10.6, Problem 14PSB

To determine

Find the length of the square using given statement.

Answer to Problem 14PSB

The side length of the square is $10\sqrt{2}$ .

Explanation of Solution

Given:

The given statement is that the square inscribed in a circle. The radius of the circle is $10$ .

Calculation:

Draw a diagram using the given statement.

  

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

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

  $PC$ is the radius of the circle.

  $PC=10$

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.

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

  $PC$ is the radius of the circle.

  $PR=2PC=2\cdot 10=20$

Let the length of the sides of the square $=x$

  $\begin{array}{l}P{R}^2=P{S}^2+S{R}^2\\ \Rightarrow {20}^2=x^2+x^2\\ \Rightarrow 400=2x^2\\ \Rightarrow \frac{400}{2}=\frac{2}{2}{x}^2\\ \Rightarrow 200=x^2\\ \Rightarrow \pm \sqrt{200}=x\\ \Rightarrow \pm 10\sqrt{2}=x\\ \Rightarrow -10\sqrt{2}=x\to \text{not}\text{valid}\\ \Rightarrow 10\sqrt{2}=x\end{array}$

Hence the side length of the square is $10\sqrt{2}$ .