Question

problem 23
clear
given: \\( \overline { a b } \cong \overline { b c } \\);
\\( \overline { b m } \\) is an altitude
of \\( \triangle a b c \\).
prove: point \\( m \\) is the
midpoint of \\( \overline { a c } \\).

1. \\( \triangle a m b \cong \triangle c m b \\) 7. hypotenuse - leg
2. 8. corresponding parts

of congruent triangles
are congruent
(c.p.c.t.c.)

1. 9.

Explanation:

Step1: Use CPCTC

Since \(\triangle AMB\cong\triangle CMB\) (by Hypotenuse - Leg), by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), we have \(AM = CM\).

Step2: Define mid - point

A mid - point of a line segment is a point that divides the line segment into two equal parts. If \(AM = CM\), then point \(M\) divides \(\overline{AC}\) into two equal parts.

Answer:

1. \(AM = CM\)
2. Statement: \(M\) is the mid - point of \(\overline{AC}\); Reason: Definition of mid - point (If a point divides a line segment into two equal parts, then it is the mid - point of that line segment)