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Any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.
Based on the characteristics of the diagram, CM is the perpendicular bisector of AB. Therefore, C is equidistant from A and B.
AC=BC
The proof can be written as a two-column proof.
Statements
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Reasons
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1. AM ≅ MB ∠ AMC and ∠ BMC are right angles |
1. Definition of a perpendicular bisector.
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2. ∠AMC≅ ∠BMC
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2. All right angles are congruent.
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3. CM≅ CM
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3. Reflexive Property of Congruence.
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4. △ ACM ≅ △ BCM
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4. SAS Congruence Theorem.
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5. AC ≅ BC
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5. Corresponding parts of congruent figures are congruent.
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6. AC = BC
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6. Definition of congruent segments.
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Using the given points A, B, C, and M as vertices, two triangles can be formed. The resulting triangles, △ ACM and △ BCM, can be proven to be congruent by identifying a congruence transformation that maps one triangle onto the other.
Since AM and BM are congruent, the distance between A and M is equal to the distance between B and M. Therefore, A is the image of B after a reflection across CM.
Since C lies on CM, a reflection across CM maps C onto itself. The same is true for M.
| Reflection Across CM | |
|---|---|
| Preimage | Image |
| B | A |
| C | C |
| M | M |
The above table shows that the images of the vertices of △ BCM are the vertices of △ ACM. Therefore, △ ACM is the image of △ BCM after a reflection across CM. Since a reflection is a rigid motion, this proves that the triangles are congruent.
Corresponding parts of congruent figures are congruent, so AC and BC are congruent. By the definition of congruent segments, AC and BC are equal. This means that C is equidistant from A and B. AC=BC The same reasoning can be applied to any point on a perpendicular bisector, showing that the point is equidistant from the endpoints of the segment.