Lenny wrote a paragraph proof of the Perpendicular Bisector Theorem. What mistake did Lenny make in his proof?

Question

HK¯¯¯¯¯¯¯¯¯
 is a perpendicular bisector of IJ¯¯¯¯¯¯
, and L is the midpoint of IJ¯¯¯¯¯¯
. M is a point on the perpendicular bisector, HK¯¯¯¯¯¯¯¯¯
. By the definition of a perpendicular bisector, I know that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯
. By the definition of a perpendicular bisector, I also know that ∠MLI
 and ∠MLJ
 are right angles. ∠MLI≅∠MLJ because of the Right Angle Congruence Theorem. I can also say that ML¯¯¯¯¯¯¯¯¯≅ML¯¯¯¯¯¯¯¯¯
 by the Reflexive Property of Congruence. With this information, I know that △MLI≅△MLJ
 by the SAS Congruence Theorem. Since the triangles are congruent, the CPCTC Theorem allows me to know that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯
. Knowing that these segments are congruent proves the Perpendicular Bisector Theorem.

(1 point)
Responses

The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯
, not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯
.
The definition of a perpendicular bisector tells you that line segment cap i liters is congruent to line segment cap j liters, not that line segment cap i cap m is congruent to line segment cap j cap m.

The SAS Congruence Theorem tells you that △MLI≅△JLM
, not that △MLI≅△MLJ
.
The SAS Congruence Theorem tells you that △MLI≅△JLM
, not that △MLI≅△MLJ
.

The definition of a perpendicular bisector tells you that∠LMI
 and ∠LMJ
 are right angles, not that ∠MLI
 and ∠MLJ
 are right angles.
The definition of a perpendicular bisector tells you that   ∠LMI
 and ∠LMJ
 are right angles, not that    ∠MLI
 and ∠MLJ
 are right angles.

Lenny did not make a mistake. His proof is correct.
Lenny did not make a mistake. His proof is correct.
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GPT-4o mini · Accepted Answer

The mistake that Lenny made in his proof is:

The definition of a perpendicular bisector tells you that line segment \( IL \) is congruent to line segment \( JL \), not that line segment \( IM \) is congruent to line segment \( JM \).

This is because being on the perpendicular bisector means that any point \( M \) on it is equidistant from the endpoints \( I \) and \( J \) (hence \( IL \cong JL \)), but it does not inherently establish congruence between the segments from \( I \) to \( M \) and from \( J \) to \( M \). Thus, Lenny's assumption about the segments \( IM \) and \( JM \) is based on a misinterpretation of the definition of a perpendicular bisector.