Question

lenny wrote a paragraph proof of the perpendicular bisector theorem. what mistake did lenny make in his proof?
\\( \overline { h k } \\) is a perpendicular bisector of \\( \overline { i j } \\), and \\( l \\) is the midpoint of \\( \overline { i j } \\). \\( m \\) is a point on the perpendicular bisector, \\( \overline { h k } \\). by the definition
of a perpendicular bisector, i know that \\( \overline { i m } \cong \overline { j m } \\). by the definition of a perpendicular bisector, i also know that \\( \angle m l i \\) and
\\( \angle m l j \\) are right angles. \\( \angle m l i \cong \angle m l j \\) because of the right angle congruence theorem. i can also say that \\( \overline { m l } \cong \overline { m l } \\) by
the reflexive property of congruence. with this information, i know that \\( \triangle m l i \cong \triangle m l j \\) by the sas congruence theorem.
since the triangles are congruent, the cpctc theorem allows me to know that \\( \overline { i l } \cong \overline { j l } \\). knowing that these segments are
congruent proves the perpendicular bisector theorem.
(1 point)
o lenny did not make a mistake his proof is correct
o the definition of a perpendicular bisector tells you that \\( \overline { i l } \cong \overline { j l } \\), not that \\( \overline { i m } \cong \overline { j m } \\).
o the definition of a perpendicular bisector tells you that \\( \angle l m i \\) and \\( \angle l m j \\) are right angles, not that \\( \angle m l i \\) and \\( \angle m l j \\) are right angles.
o the sas congruence theorem tells you that \\( \triangle m l i \cong \triangle j l m \\), not that \\( \triangle m l i \cong \triangle m l j \\)

Explanation:

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 the segment. The definition of a perpendicular bisector is a line that is perpendicular to a segment at its midpoint. So, by definition, if \( \overline{HK} \) is the perpendicular bisector of \( \overline{IJ} \) with mid - point \( L \), then \( \overline{IL}\cong\overline{JL} \) (because \( L \) is the mid - point). The statement \( \overline{IM}\cong\overline{JM} \) is what we need to prove (using congruent triangles \( \triangle MLI\) and \( \triangle MLJ\)), not what the definition of a perpendicular bisector gives us.

Answer:

The definition of a perpendicular bisector tells you that \( \overline{IL}\cong\overline{JL} \), not that \( \overline{IM}\cong\overline{JM} \).