Understanding the Perpendicular Bisector Theorem | Explained with Examples and Proofs

Converse of the Perpendicular Bisector Theorem

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

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

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

In other words, if a point P is equidistant (the same distance) from the endpoints A and B of a segment AB, then the line passing through P and perpendicular to AB is the perpendicular bisector of AB.

To understand this, let’s consider an example. Suppose we have a line segment AB, and there is a point P on the line containing AB that is equidistant from A and B.

To prove that the line passing through P and perpendicular to AB is the perpendicular bisector of AB, we need to show two things: 1) that the line is indeed perpendicular to AB, and 2) that it bisects AB.

1) To show that the line is perpendicular to AB, we can use the definition of perpendicular lines. Two lines are perpendicular if the angle formed between them measures 90 degrees. So, we need to show that the angle formed between line AP, passing through P and A, and line BP, passing through P and B, is 90 degrees. One way to prove this is by using the properties of triangles and the congruence of corresponding angles.

2) To show that the line bisects AB, we need to demonstrate that it divides AB into two congruent segments. This means that the distance from A to P is equal to the distance from B to P. We can use distance formula or geometric reasoning to prove this.

Therefore, if a point is equidistant from the endpoints of a segment, we can conclude that it lies on the perpendicular bisector of that segment. This is the converse of the Perpendicular Bisector Theorem.

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