Question

1. which is a true statement about the slopes of \\(\overline{mo}\\) and \\(\overline{oq}\\)?

a. the slope of \\(\overline{mo}\\) is greater than the slope of \\(\overline{oq}\\).
b. the slope of \\(\overline{oq}\\) is greater than the slope of \\(\overline{mo}\\).
c. the slopes are equal because \\(\frac{6 - 0}{3 - 6} = \frac{10 - 6}{6 - 8}\\).
d. the slopes are equal because \\(\frac{6 - 3}{6 - 0} = \frac{8 - 6}{10 - 6}\\).

Explanation:

To determine the correct statement about the slopes of \(\overline{MO}\) and \(\overline{OQ}\), we use the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

For option D:

• Calculate the slope of \(\overline{MO}\): Let \(M=(3, 3)\) and \(O=(6, 6)\) (from the grid). Then slope \(m_{MO}=\frac{6 - 3}{6 - 3}=\frac{3}{3} = 1\)? Wait, no, looking at the formula in D: \(\frac{6 - 3}{6 - 0}=\frac{3}{6}=\frac{1}{2}\)? Wait, no, the formula in D is \(\frac{6 - 3}{6 - 0}=\frac{3}{6}=\frac{1}{2}\) and \(\frac{8 - 6}{10 - 6}=\frac{2}{4}=\frac{1}{2}\). Wait, actually, from the grid, let's identify coordinates:
• Point \(M\): Let's assume from the grid, \(M\) is at \((3, 3)\), \(O\) at \((6, 6)\)? No, wait the dashed lines: \(M\) is at \((3, 3)\), \(O\) at \((6, 6)\)? Wait, no, the formula in D is \(\frac{6 - 3}{6 - 0}\) and \(\frac{8 - 6}{10 - 6}\). Let's compute both:
• First slope: \(\frac{6 - 3}{6 - 0}=\frac{3}{6}=\frac{1}{2}\)
• Second slope: \(\frac{8 - 6}{10 - 6}=\frac{2}{4}=\frac{1}{2}\)

So they are equal. Let's check other options:

• Option A: Says slope of \(\overline{MO}\) is greater than \(\overline{OQ}\), but if slopes are equal (from D's calculation), A is wrong.
• Option B: Says slope of \(\overline{OQ}\) is greater than \(\overline{MO}\), wrong if they are equal.
• Option C: The formula \(\frac{6 - 0}{3 - 6}=\frac{6}{-3}=-2\) and \(\frac{10 - 6}{6 - 8}=\frac{4}{-2}=-2\), but the coordinates for \(M\) and \(O\) don't match this (since \(M\) should be \((3, 3)\) and \(O\) \((6, 6)\) or similar, not \((0, 6)\) and \((3, 6)\)). So C's formula uses incorrect coordinates.
• Option D: The slopes are equal because \(\frac{6 - 3}{6 - 0}=\frac{3}{6}=\frac{1}{2}\) and \(\frac{8 - 6}{10 - 6}=\frac{2}{4}=\frac{1}{2}\), so they are equal.

Answer:

D. The slopes are equal because \(\frac{6 - 3}{6 - 0}=\frac{8 - 6}{10 - 6}\)