Understanding the Perpendicular Bisector Theorem | Explaining the Relationship between a Point, Perpendicular Bisector, and Segment

Theorem 6.1: Perpendicular Bisector Theorem

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

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

To understand this theorem, let’s break it down into its components.

1. Perpendicular Bisector: A perpendicular bisector is a line, segment, or ray that intersects another segment at a right angle (perpendicular) and divides it into two equal parts.

2. Equidistant: Equidistant means that two or more points are at the same distance from a given point, line, or plane.

The theorem states that if a point lies on a line that is the perpendicular bisector of a given segment, then that point is equidistant from the two endpoints of the segment.

To visualize this theorem, let’s consider an example. Suppose we have a line segment AB, and point P lies on the line that is the perpendicular bisector of AB. If we measure the distance from P to A and the distance from P to B, these distances will be equal.

Now, let’s understand why this theorem is true. When a line is the perpendicular bisector of a segment, it means that it divides the segment into two equal parts. Since the line is perpendicular to the segment, any point on this line will form right angles with both endpoints of the segment. This implies that the distances from the point to the two endpoints will be equal, based on the properties of right angles.

The Perpendicular Bisector Theorem is useful in various geometric proofs and constructions. It helps establish relationships between points, segments, and lines. Additionally, this theorem can be used to find the midpoint of a line segment by drawing its perpendicular bisector and locating the point of intersection.

In summary, the Perpendicular Bisector Theorem states that if a point lies on the perpendicular bisector of a segment, it will be equidistant from the endpoints of that segment.

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