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

Question

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

(1 point)
Responses

The SAS Congruence Theorem tells you that △MLI≅△JLM
△
𝑀
𝐿
𝐼
≅
△
𝐽
𝐿
𝑀
, not that △MLI≅△MLJ
△
𝑀
𝐿
𝐼
≅
△
𝑀
𝐿
𝐽
.
The SAS Congruence Theorem tells you that △MLI≅△JLM
△
𝑀
𝐿
𝐼
≅
△
𝐽
𝐿
𝑀
, not that △MLI≅△MLJ
△
𝑀
𝐿
𝐼
≅
△
𝑀
𝐿
𝐽
.

The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯
𝐼
𝐿
¯
≅
𝐽
𝐿
¯
, not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯
𝐼
𝑀
¯
≅
𝐽
𝑀
¯
.
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.

Lenny did not make a mistake. His proof is correct.
Lenny did not make a mistake. His proof is correct.

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

Answer 1

Lenny made a mistake in his proof related to the application of the properties and statements regarding the perpendicular bisector. The correct response would be:

The definition of a perpendicular bisector tells you that IL¯¯¯¯¯¯≅JL¯¯¯¯¯¯¯ 𝐼 𝐿 ¯ ≅ 𝐽 𝐿 ¯, not that IM¯¯¯¯¯¯¯¯≅JM¯¯¯¯¯¯¯¯ 𝐼 𝑀 ¯ ≅ 𝐽 𝑀 ¯.

This is incorrect because the definition of the perpendicular bisector directly states that any point on the perpendicular bisector (like point M) will be equidistant from the endpoints of the segment that it bisects (in this case, points I and J). Thus, Lenny should have directly stated that IL ≅ JL instead of assuming the congruence of IM and JM, which is not the right application in this context.