Chapter 11, Problem 39RP

To determine

To calculate:Theshaded area if AB is $12$ .

Answer to Problem 39RP

The shadedarea is $49.30$ .

Explanation of Solution

Given information:

  $\overline{AT}$ is tangent to circle with center P at point T .

AB = 12

Formula used:

Area of triangle: $A=\frac{1}{2}\ast b\ast h$

b = base of triangle

Area of sector $=\left(\frac{measureofarc}{360^o}\right)\pi r^2$

r = radius of circle.

Calculation:

  

A tangent line is perpendicular to radius drawn to the point of contact so,

ΔPTA is right angle triangle.

If AB = 12, then PB =12.

PT = 12 because it is also a radius.

ΔPTA is a ${30}^{\circ }{60}^{\circ }{90}^{\circ }\Delta$ because it has a right angle.

PT $=\frac{1}{2}\ast$ Hypotenuse $\overline{PA}$ ,

TA $=PT\sqrt{3}=12\sqrt{3}$

Shaded Area = Area of triangle - Area of Sector

Shaded Area $=\left(\frac{1}{2}b\ast h\right)-\left(\frac{measureofarc}{360^o}\right)\pi r^2$

Shaded Area $=\frac{1}{2}\left(12\sqrt{3}\right)12-\left(\frac{{60}^{\circ }}{360^o}\right)\pi {\left(12\right)}^2$

Shaded Area $=72\sqrt{3}-\frac{1}{6}144\pi$

Shaded Area $=49.30$