Question

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 the proof is incomplete.


Statement Reason
The coordinates of point D are (4, 5) and coordinates of point E are (5, 3) By the midpoint formula
Length of segment DE is Square root of 5 and length of segment AC is 2 multiplied by the square root of 5 By the distance formula
Segment DE is half the length of segment AC By substitution
Slope of segment DE is −2 and slope of segment AC is −2
Segment DE is parallel to segment AC Slopes of parallel lines are equal

Which of the following completes the proof?
A) By definition of congruence
B)Addition property of equality
C) By construction
D)By the slope formula

Answer 1

Coordinates of Point A and C are (6,8) and (8,4).

Coordinates of Point D and E are ,(4,5) and (5,3).

→Length of segment DE

Using Distance Formula

`=√((5-4)^2+(5-3)^2)\\\\=√(1+4)\\\\=√(5)`

`AC=√((8-6)^2+(8-4)^2)\\\\=√(4+16)\\\\=√(20)\\\\=2√(5)`

→So,Length of segment AC=2√5

→By substitution

`DE=(AC)/(2)`

→By the slope formula, which is

`m=(y_(2)-y_(1))/(x_(2)-x_(1))\\\\ \text{Slope of Segment DE}=(5-3)/(4-5)\\\\=-2\\\\\text{Slope of Segment AC}=(8-4)/(6-8)\\\\=(4)/(-2)\\\\=-2`

→When Slopes are equal, lines are Parallel.

So, Segment DE ║ Segment AC

The following Statement completes the proof,which is Missing

Option D: By the slope formula

Answer 2

Explanation:

Statement Reason:

1) The coordinates of point D are (4, 5) and coordinates of point E are (5, 3) By the midpoint formula

2) Length of segment DE is Square root of 5 and length of segment AC is 2 multiplied by the square root of 5 By the distance formula

3) Segment DE is half the length of segment AC By substitution

4) Slope of segment DE is -2 and slope of segment AC is -2 by the slope formula

5) Segment DE is parallel to segment AC Slopes of parallel lines are equal...