Question

proving the triangle midsegment theorem
given: d is the midpoint of \\( \overline { a b } \\); e is the midpoint of \\( \overline { a c } \\).
prove: \\( \overline { d e } \parallel \overline { b c } \\)
complete the missing parts of the paragraph proof.
proof:
to prove that \\( \overline { d e } \\) and \\( \overline { b c } \\) are parallel, we need to show that they have the same slope.
slope of \\( \overline { d e } = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } } = \frac { c - c } { a + b - b } = \\)
slope of \\( \overline { b c } = \\)
therefore, because
, \\( \overline { d e } \parallel \overline { b c } \\).

Explanation:

Step1: Calculate the slope of \( \overline{DE} \)

Given \( D(b,c) \) and \( E(a + b,c) \), using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), we have \( \frac{c - c}{a + b - b}=\frac{0}{a}=0 \)

Step2: Calculate the slope of \( \overline{BC} \)

Given \( B(0,0) \) and \( C(2a,0) \), using the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \), we have \( \frac{0 - 0}{2a-0}=0 \)

Step3: Conclude the parallelism

Since the slope of \( \overline{DE} = 0 \) and the slope of \( \overline{BC}=0 \), and two lines with the same slope are parallel.

Answer:

slope of \( \overline{DE}=0 \); slope of \( \overline{BC}=\frac{0 - 0}{2a-0}=0 \); \( \overline{DE} \parallel \overline{BC} \) because their slopes are equal (both equal to \( 0 \))