Name———————————————————————— Lesson
Lesson 11.4
11.4
Date —————————————
Study Guide
For use with the lesson “Use Geometric Probability”
goal
Use lengths and areas to find geometric probabilities.
Vocabulary
The probability of an event is a measure of the likelihood that
the event will occur.
A geometric probability is a ratio that involves a geometric
measure, such as length or area.
example 1
Use lengths to find a geometric probability
}
Find the probability that a point chosen randomly on ​AD​ is on the
given line segment. Express your answer as a fraction, a decimal,
and a percent.
A
− 20
− 15
B
− 10
−5
0
C
D
5
10
15
}
a.​AC​ 
20
}
b.​BC​ 
Solution
Length of AD​
​  
5 2 (215)
20
5 }}
​ 
   
 ​5 ​ }
 ​ 
25
10 2 (215)
} 4
The probability that the point is on AC​
​  is ​ }5 ​ , 0.8, or 80%.
}
Length of BC​
​  
}
b. P(Point is on ​BC​)  5 }}
​ 
 ​
} 
Length of AD​
​  
5 2 (25)
10
5 ​ }}
   
 ​5 }
​ 25 ​ 
10 2 (215)
} 2
The probability that a randomly chosen point is on BC​
​  is }
​ 5 ​ , 0.4, or 40%.
Exercises for Example 1
}
Find the probability that a point chosen at random on AD​
​  is on the
given line segment. Express your answer as a fraction, a decimal,
and a percent.
A
−8
−6
}
1.​AB​ 
11-54
Geometry
Chapter Resource Book
B
−4
−2
0
C
2
}
2.​BC​ 
D
4
6
8
10
12
}
3.​AC​ 
}
4.​BD​ 
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
}
Length of AC​
​  
}
a. P(Point is on ​AC​)  5 }}
​ 
 ​
} 
Name———————————————————————— Lesson
11.4
Date —————————————
Study Guide continued
For use with the lesson “Use Geometric Probability”
Use areas to find a geometric probability
Find the probability that a point chosen at random in the figure lies in the shaded region. Express your
answer as a percent.
6
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Solution
8
Step 1 Find the area of the whole rectangle, using the
formula A 5 bh.
A 5 bh 5 6(8) 5 48 square units
Step 2 Find the area of the shaded region.
The radius of the circle is one half the length of the base of the rectangle.
So, r 5 3.
The area of the semicircle is one half the area of the circle. So,
A 5 ​ }2 ​  πr 2 5 }
​ 2  ​ π p (3)2 ø 14.14 square units.
Area of shaded region 5 Area of rectangle 2 Area of semicircle
1
Lesson 11.4
example 2
1
ø 48 2 14.14
5 33.86 square units
Step 3 Find the ratio of the area of the shaded region to the total area of the figure.
  
  
 ​
P(Point lies in shaded region) 5 ​ }}
Area of shaded region
Area of total figure
33.86
5}
​  48   
​ 
ø 70.5%
The probability that a randomly chosen point lies in the shaded region is about 70.5%.
Exercises for Example 2
Find the probability that a point chosen at random in the figure lies in
the shaded region. Express the answer as a percent.
5.
6.
7.
7
1088
5 in
8 in
7
5
4
10
Geometry
Chapter Resource Book
11-55
Lesson 11.3 Areas of Regular
Polygons, continued
25. about 13.3% 26. about 6.2%
legs r and r.}Therefore, the radius of the larger
circle is r​Ï2 ​.  The ratio of the area of the smaller
Practice Level C
5
1
1. }
​ 8 ​ ; 0.625; 62.5% 2. }
​ 2 ​ ; 0.5; 50%
3
3
3. ​ } ​ ; 0.375; 37.5% 4. }
​ 4  ​; 0.75; 75% 5. 62.5%
8
π r 2
π r 2
1
circle to the larger circle is ​ }
5}
​  2   ​ 5 }
​ 2 ​ .
}   ​ 
2
6. Equilateral triangle: 1924.5 ft2; square:
2500 ft2; regular pentagon: 2752.8 ft2; regular
hexagon: 2886.8 ft2; circle: 3183.1 ft2; The farmer
should choose a circle.
9. about 17.3% 10. about 20.3% 11. 75%
12. 50% 13. 25% 14. 56.25% 15. 37.5%
Lesson 11.4 Use Geometric
Probability
16. 93.75% 17. 35 18. about 19.9%
Teaching Guide
22. about 31.83% 23. about 75.68%
1. Answers may vary but most students will pick
sector A because it is most likely to occur.
2. Answers may vary but most students will pick
sector D because it is least likely to occur.
5
1
3. }
​ 6 ​ 4. }
​ 24  ​
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
6. 68.75% 7. 62.5% 8. about 21.46%
Practice Level A
3
1
1. }
​ 2 ​ ; 0.5; 50% 2. }
​ 8 ​ ; 0.375; 37.5%
3
1
3. }
​ 4  ​; 0.25; 25% 4. ​ }4  ​; 0.75; 75% 5. 50%
6. about 21.46% 7. about 36.3% 8. 80%
9. 75% 10. about 17.65% 11. 50% 12. 25%
13. 37.5% 14. 75% 15. 70 square units
16. about 39.8% 17. about 60.2% 18. 25%
19. 25% 20. about 47.64% 21. 25% 22. 32%
23. C 24. 25% 25. about 38.9%
26. about 8.3% 27. about 19.4%
28. about 8.3% 29. 10% 30. about 6.4%
Practice Level B
3
5
1. }
​ 8 ​; 0.375; 37.5% 2. ​ }8 ​; 0.625; 62.5%
3
1
3. }
​ 2 ​; 0.5; 50% 4. ​ }4 ​; 0.75; 75% 5. about 47.6%
6. 68.75% 7. 25% 8. 25% 9. about 56.95%
10. about 17.3% 11. 12.5% 12. 25%
13. 62.5% 14. 0% 15. 49.5 16. about 28.1%
17. about 71.9% 18. about 44.2%
19. about 16.7% 20. about 16.7%
21. about 78.5% 22. 32% 23. 50% 24. C
answers
π r 2
π (r​Ï2 ​ )
27. about 4.5% 28. about 29.8%
19. about 80.1% 20. 47.5% 21. 47.5%
24. about 57.8% 25. 20% 26. 25%
27. a. 6.25% b. 18.75% c. 31.25% d. 43.75%
Study Guide
3
1
1. }
​ 8 ​ 5 0.375 5 37.5% 2. }
​   ​  5 0.125 5 12.5%
8
5
1
3. }
​ 2 ​ 5 0.5 5 50% 4. }
​ 8 ​ 5 0.625 5 62.5%
5. 18.75% 6. 25.37% 7. 45%
Problem Solving Workshop:
Mixed Problem Solving
1. a. 20% b. 80% 2. a. 15 in., 10 in.
b. about 47.12 in., about 31.42 in. c. about 8.49
revolutions, about 12.73 revolutions
3. a. about 1.3% b. 4; The probability of a
1
marker landing in the target region is }
​ 75  ​ . Solve
x
1
the proportion }
​ 300   ​ 5 }
​ 75  ​ to find the number x of
markers out of 300 that you would expect to land
in the target region. 4. about 46
5. 132.73 in.2, 40.84 in.; 530.93 in.2, 81.68 in.;
Doubling the radius quadruples the area of the
circle because you are substituting 2r for r in the
equation for the area of a circle. Doubling the
radius doubles the circumference because you
are substituting 2r for r in the equation for the
­circumference of a circle. 6. Answers will vary.
7. a. about 494.8 mi b. 9.63 mi; The distance
the boat travels in the beam of light is a chord
of the circle. The chord that is the perpendicular
bisector of this chord is the diameter of the circle.
So, you can find the center of the circle, which
is the lighthouse. The distance from the center
along the diameter to the chord is the shortest distance because the shortest distance between a
Geometry
Chapter Resource Book
A61