Geometry Essentials for Dummies by Mark Ryan

Author:Mark Ryan [Ryan, Mark]
Language: eng
Format: epub
Tags: Mathematics, Geometry, General
ISBN: 9781118095799
Google: hABb-bvL6JoC
Publisher: John Wiley & Sons
Published: 2011-01-01T18:30:00+00:00

The Two Equidistance Theorems

Although congruent triangles are the focus of this chapter, in this section, I give you two theorems that you can often use instead of proving triangles congruent. Even though you see congruent triangles in this section’s proof diagrams, you don’t have to prove the triangles congruent; one of the equidistance theorems gives you a shortcut to the prove statement.

When doing triangle proofs, be alert for two possibilities: Look for congruent triangles and think about ways to prove them congruent, but at the same time, try to see whether one of the equidistance theorems can get you around the congruent triangle issue.

Determining a perpendicular bisector

The first equidistance theorem tells you that two points determine the perpendicular bisector of a segment. (To “determine” something means to lock in its position, basically to show you where something is.)

Two equidistant points determine the perpendicular bisector: If two points are each (one at a time) equidistant from the endpoints of a segment, then those points determine the perpendicular bisector of the segment. (Here’s an easy way to think about it: If you have two pairs of congruent segments, then there’s a perpendicular bisector.)

The best way to understand this royal mouthful is visually. Consider the kite-shaped diagram in Figure 5-9.

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