Question

given: d is the midpoint of \\(\overline{ab}\\); e is the midpoint of \\(\overline{ac}\\). prove: \\(\overline{de} \parallel \overline{bc}\\)
diagram: coordinate plane with points b(0,0), c(2a,0) on x-axis; a(2b,2c); d(b,c) (midpoint of ab); e(a+b,c) (midpoint of ac); segment de connects d and e, segment bc connects b and c
complete the missing parts of the paragraph proof.
proof:
to prove that \\(\overline{de}\\) and \\(\overline{bc}\\) are parallel, we need to show that they have the same slope.
slope of \\(\overline{de} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{c - c}{a + b - b} = \square\\)
slope of \\(\overline{bc} = \square\\)
therefore, bec blue box the slopes are \\(\

$$\begin{matrix}2a\\\\0\\\\undefined\\end{matrix}$$

\\), \\(\overline{de} \parallel \overline{bc}\\).

Explanation:

Step1: Calculate slope of DE

The formula for slope is $\frac{y_2 - y_1}{x_2 - x_1}$. For points D$(b, c)$ and E$(a + b, c)$, $y_2 - y_1 = c - c = 0$ and $x_2 - x_1 = (a + b) - b = a$. So slope of DE is $\frac{0}{a} = 0$.

Step2: Calculate slope of BC

For points B$(0, 0)$ and C$(2a, 0)$, using the slope formula $\frac{y_2 - y_1}{x_2 - x_1}$, we have $y_2 - y_1 = 0 - 0 = 0$ and $x_2 - x_1 = 2a - 0 = 2a$. So slope of BC is $\frac{0}{2a} = 0$.

Step3: Compare slopes

Since the slope of DE is 0 and the slope of BC is 0, the slopes are equal. When two lines have equal slopes, they are parallel.

Answer:

Slope of $\overline{DE}$: $0$; Slope of $\overline{BC}$: $0$; Because the slopes are equal, $\overline{DE} \parallel \overline{BC}$.