Question

select the correct answer.
quad is a quadrilateral with vertices q(-3,2), u(3,0), a(6,-5), and d(0,-3). the slope for \\(\overline{qu}\\) is \\(\frac{0 - 2}{3 - (-3)} = -\frac{1}{3}\\). the slope for \\(\overline{ua}\\) is \\(\frac{-5 - 0}{6 - 3} = -\frac{5}{3}\\). the slope for \\(\overline{ad}\\) is \\(\frac{-3 - (-5)}{0 - 6} = -\frac{1}{3}\\). the slope for \\(\overline{dq}\\) is \\(\frac{-3 - 2}{0 - (-3)} = -\frac{5}{3}\\). so, ______. therefore, quad is a parallelogram.
what is the missing step in the proof?
a. \\(\overline{qu} \parallel \overline{ad}\\) and \\(\overline{ua} \parallel \overline{dq}\\) because the product of the slopes is -1.
b. \\(\overline{qu} \perp \overline{ad}\\) and \\(\overline{ua} \perp \overline{dq}\\) because the segments have the same slope.
c. \\(\overline{qu} \perp \overline{ad}\\) and \\(\overline{ua} \perp \overline{dq}\\) because the product of the slopes is -1.
d. \\(\overline{qu} \parallel \overline{ad}\\) and \\(\overline{ua} \parallel \overline{dq}\\) because the segments have the same slope.

Explanation:

A quadrilateral is a parallelogram if both pairs of opposite sides are parallel. Parallel lines have identical slopes. From the given calculations:

• Slope of $\overline{QU} = -\frac{1}{3}$, slope of $\overline{AD} = -\frac{1}{3}$ (equal slopes, so parallel)
• Slope of $\overline{UA} = -\frac{5}{3}$, slope of $\overline{DQ} = -\frac{5}{3}$ (equal slopes, so parallel)

Options A and C refer to perpendicular lines (product of slopes = -1), which is not the case here. Option B incorrectly states equal slopes mean perpendicular. Only option D correctly links equal slopes to parallel opposite sides.

Answer:

D. $\overline{QU} \parallel \overline{AD}$ and $\overline{UA} \parallel \overline{DQ}$ because the segments have the same slope.