Title: Example 1 Geometric Mean
Author: Tim Heppner
Last saved by: MHE
Company: MCGRAW-HILL

Geometry	Chapter 11

Lesson 11-5

Example 1  Probability with Area
A square game board has black and white regions
as shown. What is the chance that a dart thrown at
the board will land on a black region of the board?

You want to find the probability of landing on a black area, not a white area.

We need to divide the area of the black regions by 	
the total area of the game board. Divide the square
into 36 small units as shown.
The black regions have an area of 16 square units.
The total area is 36 square units.
The probability of throwing a dart onto a black region
of the board is 16/36 or 4/9.

Example 2  Probability with Sectors
a.	Find the area of the blue sector.	
	Use the formula to find the area of the sector.
	A	= N/360πr^2	Area of a sector
		= 63/360π(10^2)	N = 63, r = 10
		= 17.5π	Simplify.

b.	Find the probability that a point chosen at random lies in the blue region.
	To find the probability, divide the area of the sector by the area of the circle. 
	The area of the circle is πr^2 with a radius of 10.
	P(blue)	= (area of sector)/(area of circle)	Geometric probability formula
		= 	Area of sector = 17.5π, area of circle = π (10^2)	
		= 0.175	Simplify.

	The probability that a random point is in the blue sector is about 0.175 or 17.5%.

Example 3  Probability with Segments
A square is inscribed in a circle with a diameter of 26.
a.	Find the area of the blue segments.
	
Area of three congruent sectors:
	A	= ( N/360πr^2 ) 3	(Area of a sector · 3	
		= [ 90/360π(13^2)] 3	N = 90, r = 13
		= 126.75π	Simplify.
		≈ 398.20	Use a calculator.

	Area of the three congruent triangles:
	Since the square was inscribed in the circle, each triangle is a 45°-45°-90° 
	triangle. The legs of the triangles are radii of the circle, or 13 units. 
	So, the base and the height of each triangle is 13.

	Next, use the formula for the area of a triangle.
	A	= (1/2bh 3	(Area of a triangle · 3
		= [ 1/2(13)(13) ] 3	b = h = 13
		= 253.5	Simplify.

	Area of three congruent segments:
	area of three congruent segments	= area of sectors- area of triangles
			≈ 398.20 - 253.5
			≈ 144.7

b.	Find the probability that a point chosen at random lies in the blue region.
	Divide the area of the three segments by the area of the circle to find the probability. First, find the area of the circle. The radius is 13, so the area is π(13^2) or about 530.93 square units.
	P(blue)	= (area of three segments)/(area of circle)
		≈ 
		≈ 0.27

	The probability that a random point is on the area represented by the three blue segments is about 0.27 or 27%.