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GEOMETRY The Triangle Inequality

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Presentation on theme: "GEOMETRY The Triangle Inequality"— Presentation transcript:

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

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