1. Lesson 7.5
Triangle Inequality pp

2. Objectives: 1. To state and prove the triangle inequality.
2. To apply the triangle inequality to triangle existence problems and to problem-solving strategies.

3. A A 15 12 9 7 5 A 3 B 8 C B 8 C B 8 C

4. Definition
A point M is Between A and B if AM + MB = AB. The correct notation is A-M-B.

5. Theorem 7.14
Triangle Inequality. The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

6. 2 3 ? 7

7. EXAMPLE In a business office the equipment must be arranged carefully for efficient use. Becky uses primarily the computer on her desk, the photocopier, and the fax machine. A triangular arrangement is desired to facilitate access between all three machines. The distance between her desk and the photocopier must be the shortest, and the total distance between the three items should not exceed 20 feet.

8. Comp.-Copier Copier-fax Comp.-Fax
1. 5 ft 7 ft 6 ft
2. 3 ft 6 ft 9 ft

9. Practice: Can a triangle be. constructed with segments. of
Practice: Can a triangle be constructed with segments of length 5, 8, and 13?
1. Yes
2. No

10. Practice: Can a triangle be. constructed with segments
Practice: Can a triangle be constructed with segments of length 4, 6, and 1?
1. Yes
2. No

11. Practice: Given two sides of a. triangle equal 4 and 7, what
Practice: Given two sides of a triangle equal 4 and 7, what is the range of possible values for the third side?
3  x  11
The third side must be greater than the difference but less than the sum.

12. Practice: Given two sides of a. triangle equal 9 and 18,
Practice: Given two sides of a triangle equal 9 and 18, which of the following is a possible length for the third side?
1. 9
2. 28
3. 12
4. 8

13. Homework pp

14. ►A. Exercises
State whether triangles with the following side lengths exist. If not, state the reason.
a b c

15. ►A. Exercises 7. AB = 3, BC = 5, AC = 2
AB, BC, and AC are given. Make a sketch and identify the type of triangle. If none is formed, explain why (impossible, collinear).
7. AB = 3, BC = 5, AC = 2

16. ►A. Exercises 9. AB = 5, BC = 3, AC = 1
AB, BC, and AC are given. Make a sketch and identify the type of triangle. If none is formed, explain why (impossible, collinear).
9. AB = 5, BC = 3, AC = 1

17. ►A. Exercises 11. AB = 5, BC = 3, AC = 3
AB, BC, and AC are given. Make a sketch and identify the type of triangle. If none is formed, explain why (impossible, collinear).
11. AB = 5, BC = 3, AC = 3

18. ►B. Exercises
Refer to Becky’s office in the example in this section. Are the following measurements possible? If not, explain why.
Computer-Copier Copier-Fax Computer-Fax
13. 5 feet 7 feet 8 feet
15. 6 feet 12 feet 10 feet
17. 5 feet 9 feet 6 feet

19. ■ Cumulative Review
Supply the missing reasons needed to complete the following proof.
Given: AD  AB, CB  CD
Prove: D  B A D x C B

20. STATEMENTS REASONS
22. ADAB; CBCD 22.
23. A and C are 23.
right angles
24. A  C 24.
25. AXD  CXD 25.
26. D  B 26.
Given
Def. of Perp.
All rt. ’s are 
Vert.  Thm.
Third ’s 