The Perpendicular Bisector Theorem: Explained With Proofs And Applications In Geometry And Trigonometry

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.If CP is the perpendicular bisector of AB, then AC = BC

The Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. In other words, if a line passes through the midpoint of a line segment at a 90-degree angle to that segment, then that line divides the original segment into two equal halves.

This theorem is useful in geometry and trigonometry as it can help find the distance between two points or the location of a point if its distance from two other points is known. It is also used in constructions, such as creating perpendicular lines or bisecting angles.

To prove this theorem, you can use the Pythagorean Theorem and the definition of perpendicular lines. Let’s say we have a line segment AB, with midpoint M.

1. Draw a line passing through M that is perpendicular to AB. Let’s call this line PQ.

2. Using the Pythagorean Theorem, we can express the distance from A to M (AM) and the distance from B to M (BM) in terms of AB and MP:

AM^2 = MP^2 + AP^2
BM^2 = MP^2 + BP^2

3. Since PQ is perpendicular to AB, AP = BP. Therefore, we can set these equations equal to each other:

MP^2 + AP^2 = MP^2 + BP^2
AP^2 = BP^2
AP = BP

4. This proves that any point on the perpendicular bisector of AB is equidistant from A and B.

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