Chapter 11, Problem 45RP

To determine

To calculate: The area of the circle containing arc AB .

Answer to Problem 45RP

The area of the circle containing arc AB is $106.8$ .

Explanation of Solution

Given information:

  $m\widehat{AB}={90}^{\circ }$

Formula used:

The distance between two points by using the distance formula, which is an application of the Pythagoras theorem.

D $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Area of a circle: $A=\pi r^2$

r = radius of circle

Calculation:

  

The length of chord $\overline{AB}$ can be calculated by applying distance formula.

  $\begin{array}{l}AB=\sqrt{{\left(11-3\right)}^2+{\left(5-7\right)}^2}\\ =\sqrt{{\left(8\right)}^2+{\left(-2\right)}^2}\\ =\sqrt{64+4}\\ =\sqrt{68}\end{array}$

Since $m\widehat{AB}={90}^{\circ }$ , the chord is also one side of an inscribed square, the diagonals of which intersect at the circle’s center. The radius of the circle is half the length of a diagonal, or $\frac{\sqrt{136}}{2}$ .

Thus,

Area of a circle: $A=\pi {\left(\frac{\sqrt{136}}{2}\right)}^2=\frac{136\pi }{4}=34\pi \approx 106.8$