1. GEOMETRY 5.4 The Triangle Inequality
Objectives:
State and apply the Triangle Inequality Theorem
Determine the shortest distance between a
a point and a line

2. GEOMETRY 5.4

3. GEOMETRY 5.4

4. GEOMETRY 5.4

5. GEOMETRY 5.4
GEOMETRY 5.4

6. GEOMETRY 5.4
GEOMETRY 5.4
To Determine whether 3 lengths could make a Triangle:

7. GEOMETRY 5.4
GEOMETRY 5.4
To Determine whether 3 lengths could make a Triangle:
ADD the Two Smallest Lengths (a + b)

8. GEOMETRY 5.4
GEOMETRY 5.4
To Determine whether 3 lengths could make a Triangle:
ADD the Two Smallest Lengths (a + b)
Compare the SUM to the Longest Length (c)

9. GEOMETRY 5.4
GEOMETRY 5.4
To Determine whether 3 lengths could make a Triangle:
ADD the Two Smallest Lengths (a + b)
Compare the SUM to the Longest Length (c)
To make a Triangle: a + b > c

10. GEOMETRY 5.4

11. GEOMETRY 5.4

12. GEOMETRY 5.4

13. GEOMETRY 5.4

14. x > b – a a + b > x To determine the RANGE of possible Lengths
of the 3rd side of a Triangle:
Given length a and b, with a < b
Find the RANGE of the third side, x
Lower Limit: Upper Limit:
x > b – a a + b > x

15. GEOMETRY 5.4

16. GEOMETRY 5.4
REMEMBER:
Distance between 2 points on the graphing plane:

17. GEOMETRY 5.4

18. GEOMETRY 5.4

19. GEOMETRY 5.4

20. GEOMETRY 5.4
AB + BC > AC
BC + AC > AB
AC + AB > BC

21. GEOMETRY 5.4
GEOMETRY 5.4
PQ + QR > PR
QR + PR > PQ
PR + PQ > QR

22. GEOMETRY 5.4 R O
Given: Triangle ROS
Prove: SO + OR > RS S

23. GEOMETRY 5.4
GEOMETRY 5.4 R O
Given: Triangle ROS T
Prove: SO + OR > RS S
Start: Draw segment OT
so that OT = SO

24. GEOMETRY 5.4 D
Given:
Prove: BD + DC > AC 1 2 A B C

25. GEOMETRY 5.4
Given: D
Prove: BE + ED + AC > DC E 1 2 A B C

26. GEOMETRY 5.4

27. P
Which Segment from P is the Shortest?

28. SHORTEST Segment Theorem
The PERPENDICULAR segment from a Point to a Line IS
the SHORTEST Segment from the Point to the Line.

29. SHORTEST Segment Theorem
The PERPENDICULAR segment from a Point to a Line IS
the SHORTEST Segment from the Point to the Line.
Given: P
Prove: 2 1 j Q A

30. SHORTEST Segment Theorem
A Corollary
The LONGEST Side of a Right Triangle is the
HYPOTENUSE.

31. SHORTEST Segment Theorem
A Corollary
The LONGEST Side of a Right Triangle is the
HYPOTENUSE.
Another Corollary
The Perpendicular Segment from a Point to a Plane is the
SHORTEST Segment from the Point to the Plane.

32. GEOMETRY 5.4