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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
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GEOMETRY 5.4
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GEOMETRY 5.4
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GEOMETRY 5.4
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GEOMETRY 5.4 GEOMETRY 5.4
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GEOMETRY 5.4 GEOMETRY 5.4 To Determine whether 3 lengths could make a Triangle:
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GEOMETRY 5.4 GEOMETRY 5.4 To Determine whether 3 lengths could make a Triangle: ADD the Two Smallest Lengths (a + b)
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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)
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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
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GEOMETRY 5.4
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GEOMETRY 5.4
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GEOMETRY 5.4
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GEOMETRY 5.4
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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
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GEOMETRY 5.4
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GEOMETRY 5.4 REMEMBER: Distance between 2 points on the graphing plane:
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GEOMETRY 5.4
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GEOMETRY 5.4
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GEOMETRY 5.4
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GEOMETRY 5.4 AB + BC > AC BC + AC > AB AC + AB > BC
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GEOMETRY 5.4 GEOMETRY 5.4 PQ + QR > PR QR + PR > PQ PR + PQ > QR
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GEOMETRY 5.4 R O Given: Triangle ROS Prove: SO + OR > RS S
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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
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GEOMETRY 5.4 D Given: Prove: BD + DC > AC 1 2 A B C
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GEOMETRY 5.4 Given: D Prove: BE + ED + AC > DC E 1 2 A B C
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GEOMETRY 5.4
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P Which Segment from P is the Shortest?
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SHORTEST Segment Theorem
The PERPENDICULAR segment from a Point to a Line IS the SHORTEST Segment from the Point to the Line.
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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
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SHORTEST Segment Theorem
A Corollary The LONGEST Side of a Right Triangle is the HYPOTENUSE.
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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.
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GEOMETRY 5.4
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