Theorem: The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. A two-column proof of the theorem is shown, but a justification is missing.

\begin{tabular}{|l|l|}
\hline
Statement & Justification \\
\hline
The coordinates of point [tex][tex]$D$[/tex][/tex] are [tex][tex]$(4,5)$[/tex][/tex] and the coordinates of point [tex][tex]$E$[/tex][/tex] are [tex][tex]$(5,3)$[/tex][/tex] & \\
\hline
Length of segment [tex][tex]$DE$[/tex][/tex] is [tex][tex]$\sqrt{5}$[/tex][/tex] and the length of segment [tex][tex]$AC$[/tex][/tex] is [tex][tex]$2 \sqrt{5}$[/tex][/tex] & Distance Formula \\
\hline
Segment [tex][tex]$DE$[/tex][/tex] is half the length of segment [tex][tex]$AC$[/tex][/tex]. & Substitution Property of Equality \\
\hline
Slope of segment [tex][tex]$DE$[/tex][/tex] is -2 and the slope of segment [tex][tex]$AC$[/tex][/tex] is -2. & Slope Formula \\
\hline
Segment [tex][tex]$DE$[/tex][/tex] is parallel to segment [tex][tex]$AC$[/tex][/tex]. & Slopes of parallel lines are equal. \\
\hline
\end{tabular}

Which is the missing justification?



Answer :

To find the missing justification, we need to determine how the coordinates of points [tex]\( D \)[/tex] and [tex]\( E \)[/tex] are calculated. Here’s the step-by-step solution:

1. Identify the midpoints:

- Let's assume point [tex]\( A \)[/tex] has coordinates [tex]\((2, 2)\)[/tex] and point [tex]\( B \)[/tex] has coordinates [tex]\((6, 8)\)[/tex].
- The midpoint [tex]\( D \)[/tex] of segment [tex]\( AB \)[/tex] is calculated using the midpoint formula:
[tex]\[ D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
- Substituting [tex]\((x_1, y_1) = (2, 2)\)[/tex] and [tex]\((x_2, y_2) = (6, 8)\)[/tex]:
[tex]\[ D = \left( \frac{2 + 6}{2}, \frac{2 + 8}{2} \right) = \left( \frac{8}{2}, \frac{10}{2} \right) = (4, 5) \][/tex]

- Now, let's assume point [tex]\( C \)[/tex] has coordinates [tex]\((8, 4)\)[/tex].
- The midpoint [tex]\( E \)[/tex] of segment [tex]\( BC \)[/tex] is calculated similarly:
[tex]\[ E = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) \][/tex]
- Substituting [tex]\((x_2, y_2) = (6, 8)\)[/tex] and [tex]\((x_3, y_3) = (8, 4)\)[/tex]:
[tex]\[ E = \left( \frac{6 + 8}{2}, \frac{8 + 4}{2} \right) = \left( \frac{14}{2}, \frac{12}{2} \right) = (7, 6) \][/tex]

Therefore, the missing justification is:

The coordinates of point [tex]\( D \)[/tex] are [tex]\((4, 5)\)[/tex] and coordinates of point [tex]\( E \)[/tex] are [tex]\((7, 6)\)[/tex] because they are the midpoints of segments [tex]\( AB \)[/tex] and [tex]\( BC \)[/tex] respectively, using the midpoint formula.