Question

given: d is the midpoint of \overline{ab}; e is the midpoint of \overline{ac}. prove: \overline{de} \parallel \overline{bc}
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, because \square, \overline{de} \parallel \overline{bc}.
options for last dropdown: the slopes are equal, the slopes are not equal, the slopes are negative reciprocals
diagram: coordinate plane with b(0,0), c(2a,0), a(2b,2c), d(b,c), e(a+b,c)

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$(a + b, c)$, substitute into the formula: $\frac{c - c}{a + b - b}$. Simplify numerator and denominator: numerator $c - c = 0$, denominator $a + b - b = a$. So slope of DE is $\frac{0}{a}=0$.

Step2: Calculate slope of BC

For points B$(0, 0)$ and C$(2a, 0)$, use slope formula $\frac{y_2 - y_1}{x_2 - x_1}$. Substitute: $\frac{0 - 0}{2a - 0}=\frac{0}{2a}=0$.

Step3: Determine parallelism

Two lines are parallel if their slopes are equal. Since slope of DE and slope of BC are both 0, they are equal.

Answer:

Slope of $\overline{DE}$: $0$; Slope of $\overline{BC}$: $0$; Reason: the slopes are equal