Find the length of the segment. Then find the coordinates of the midpoint of the segment.
20. 24 22 2123 0 1 2 3 4 5 6 7
21. 29 27 26 25 24 23 22 2128 210
The endpoints of two segments are given. Find each segment length. Tell whether the segments are congruent.
22. }
AB : A(2, 6), B(0, 3) 23. }
RS : R(5, 4), S(0, 4)
}
CD : C(21, 0), D(1, 3) } TU : T(24, 23), U(21, 1)
24. } KL : K(24, 13), L(210, 6) 25. }
OP : O(6, 22), P(3, 22)
} MN : M(21,22), N(21, 211) }
QR : Q(5, 2), R(1, 5)
26. Distances Your house and the mall are 9.6 miles apart on the same straight road. The movie theater is halfway between your house and the mall, on the same road.
a. Draw and label a sketch to represent this situation. How far is your house from the movie theater?
b. You walk at an average speed of 3.2 miles per hour. About how long would it take you to walk to the movie theater from your house?
In Exercises 27–29, use the map. The locations of the towns on the map are: Dunkirk (0, 0), Clearfield (10, 2), Lake City (5, 7), and Allentown (1, 4). The coordinates are given in miles.
27. Find the distance between each pair of towns.
1
1
Dunkirk (0, 0)
Allentown (1, 4)
Lake City (5, 7)
Clearfield (10, 2)
Distance (mi)
Distance (mi)
Round to the nearest tenth of a mile.
28. Which two towns are closest together? Which two towns are farthest apart?
29. The map is being used to plan a 26-mile marathon. Which of the following plans is the best route for the marathon? Explain.
A. Dunkirk to Clearfield to Allentown to Dunkirk
B. Dunkirk to Clearfield to Lake City to Allentown to Dunkirk
C. Dunkirk to Lake City to Clearfield to Dunkirk
D. Dunkirk to Lake City to Allentown to Dunkirk
Practice B continuedFor use with the lesson “Use Midpoint and Distance Formulas”
5. Answers will vary. 6. Repeat Exercises 124. Use a straight edge to draw
} CD . Let P be the point
where }
CD and }
AB intersect. nAPC and nBPC are right triangles.
Challenge Practice
1. By the Segment Addition Postulate, you know AB 1 BC 5 AC and BC 1 CD 5 BD. Because AB 5 CD, substitute CD for AB in AB 1 BC 5 AC to obtain CD 1 BC 5 AC. You can then conclude AC 5 BD. 2.
} QS >
} SU > } TV > } RT ;
} QT >
} SV ;
} QU > } RV
3. Not sufficient. Counterexample:
A D F B C
4. Not sufficient. Counterexample:
AD F CB
5. Sufficient. A D F C BAD 1 DF 1 FC 1 CB 5 AB
6. a. AF 5 GE 5 CH 5 HI 5 ID 5 4; HD 5 IC 5 FB 5 DG 5 8; AC 5 CE 5 6;
AB 5 CB 5 CD 5 DE 5 12 b. 4 }
13 ; There are a
total of 13 segments in the diagram and 4 of those segments have lengths greater than 8.
7. x2 1 x 5 12; x 5 3; LM 5 9, MN 5 3
8. x2 2 5x 5 50; x 5 10; LM 5 40, MN 5 10
9. 2x2 1 9x 5 56; x 5 7 }
2 ; LM 5
49 }
4 , MN 5
175 }
4
Lesson Use Midpoint and Distance Formulas
Teaching Guide
1–4. Answers will vary.
5. Sample answer: No. The Distance Formula was not needed if the segment connecting two points was horizontal or vertical. Counting or subtraction easily determined the answer.
6. Sample answer: Estimation and comparison determined the distances to try first.
Technology Activity
1.
x
y
2
1
A
MB
AM 5 5.00BM 5 5.00
M : (21, 22)
2. They are congruent. 3. It is the average of the x- and y-coordinates of points A and B.
4. The x-coordinate of the midpoint is the average of the x-coordinates of the endpoints. The y-coordinate of the midpoint is the average of the y-coordinates of the endpoints.
Lesson Use Midpoint and Distance Formulas, continued22. AB 5 Ï
}
13 ; CD 5 Ï}
13 ; congruent
23. RS 5 5; TU 5 5; congruent
24. KL 5 Ï}
85 ; MN 5 9; not congruent
25. OP 5 3; QR 5 5; not congruent
26. a. Movie Theater MallHouse
9.6 miles
4.8 mi b. 1.5 h
27. Dunkirk to Clearfield 5 10.2 mi; Dunkirk to Lake City 5 8.6 mi; Dunkirk to Allentown 5 4.1 mi; Clearfield to Lake City 5 7.1 mi; Clearfield to Allentown 5 9.2 mi; Lake City to Allentown 5 5 mi 28. Dunkirk and Allentown; Dunkirk and Clearfield 29. Choice C; the total distance of the path is closest to 26 mi.
Practice Level C
1. 23 cm 2. 19 1 }
4 in. 3. 29
1 }
6 ft 4. 8.45 m
5. 7 6. 142 7. 24 2 }
3 8. (8, 1) 9. (2.5, 4)
10. (25.5, 21) 11. (211.5, 22.5)
12. (0, 210) 13. (211, 23) 14. (22, 22)
15. (218, 23) 16. 6.3 17. 7.8 18. 8.1
19. 6.7 20. 16; 26 21. 45; 27.5
22. AB < 8.6, CD < 8.5; not congruent
23. RS 5 10, TU 5 10; congruent
24. KL < 12.0, MN < 12.0; congruent
25. OP < 7.8, QR < 8.1; not congruent
26. 3737 units 27. 5458 units
28. 5296 units 29. 3996 units
30. Dunkirk to Clearfield: 10.8 mi; Dunkirk to Lake City: 14.4 mi; Dunkirk to Allentown: 12.6 mi; Clearfield to Lake City: 4.5 mi; Clearfield to Allentown: 10 mi; Lake City to Allentown: 8.9 mi 31. Clearfield and Lake City; Dunkirk and Lake City 32. D; The total distance of this route (35.9 mi) is closest to 36 miles.
Study Guide
1. 20 cm 2. 50 mm 3. 74 in. 4. 15 cm
5. (2, 4) 6. 1 4, 3 }
2 2 7. (8, 3) 8. (3, 2)
9. (1, 1) 10. 1 5 } 2 , 2
5 } 2 2 11. 7.2 12. 6.4
Problem Solving Workshop: Mixed Problem Solving
1. a. 14 mi b. about 11.4 mi c. about 2.6 mi
2. a. A B C b. 1125 ft 3. Y(2, 21); Use the points S and T to find point W. Then use the midpoint formula with point W and point X to find the coordinates of point Y. 4. 2 5. EDC, DCF, CFE, FED; The intersection of planes ABC and BGF is @##$ BC . 6. a. 82 mi b. about 1.5 h c. Yes; The total time of the trip is 1.5 1 3(1.75) 5 6.75 hours which is less than 8 hours.
Challenge Practice
1. 1 3x1 1 x2 }
4 ,
3y1 1 y2 }
4 2 , 1 x1 1 x2
} 2 ,
y1 1 y2 }
2 2 ,
1 x1 1 3x2 }
4 ,
y1 1 3y2 }
4 2 2. a. 1 7 }
4 , 2
7 } 4 2 , 1 5 }
2 , 2
3 } 2 2 ,
1 13 }
4 , 2
5 } 4 2 b. 1 2
3 } 2 , 2
9 } 4 2 , 1 21, 2
3 } 2 2 , 1 2
1 } 2 , 2
3 } 4 2
3. Sample answer: To determine whether three points A, B, and C in a coordinate plane are collinear, find AB, BC, and AC. By the Segment Addition Postulate, if AB 1 BC 5 AC, then B is between A and C, and when one point is between two other points, then the three points are collinear.
4. Collinear 5. Not collinear 6. Not collinear
7. Collinear 8. AB 5 Ï}
29 , M 5 1 5, 3, 13
} 2 2
9. AB 5 Ï}
13 , M 5 1 3, 1, 15
} 2 2
10. AB 5 Ï}
19 , M 5 1 2 9 } 2 ,
9 }
2 ,
13 }
2 2
11. AB 5 17, M 5 1 5 } 2 , 2, 6 2
12. a. (15, 22.5), (45, 22.5) b. 30 units
Lesson Measure and Classify Angles
Teaching Guide
1. piece 1; Sample answer: no, that piece is so large that there wouldn’t be room for seven of the other pieces within the circle.
