The Perpendicular Bisector Theorem In Geometry: An Easy-To-Understand Explanation And Its Applications

Perpendicular Bisector Theorem

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

The Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment.

In simpler terms, this means that if we draw a line segment and locate its midpoint, and then draw a line perpendicular to the line segment so that it passes through the midpoint, then any point on that perpendicular line will be equidistant from the two endpoints of the original line segment.

For example, consider a line segment AB. If we draw a line perpendicular to AB that passes through its midpoint M, any point P on that perpendicular line will be equidistant from A and B. This means that the distance from P to A will be the same as the distance from P to B.

This property is useful in geometry, particularly in constructing geometric figures and proving theorems. For instance, we can use the perpendicular bisector theorem to construct an equilateral triangle whose vertices lie on a given line segment. We simply draw the perpendicular bisector of the line segment and then locate two points on that perpendicular line which are equidistant from the endpoints of the segment. The third vertex of the equilateral triangle is then located at the intersection of the perpendicular bisector and a circle centered at either of the two equidistant points with a radius equal to the length of the line segment.

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