Question

a line and two triangles are shown on the coordinate plane.
which statements are true? select all that apply.
the slope of \\( \overline { p r } \\) is equal to the slope of \\( \overline { q s } \\).
the slope of \\( \overline { p r } \\) is equal to the slope of \\( \overline { p s } \\).
the slope of \\( \overline { p r } \\) is equal to the slope of the line.
the slope of the line is equal to \\( \frac { t r } { u s } \\).
the slope of the line is equal to \\( \frac { p t } { t r } \\).
the slope of the line is equal to \\( \frac { q u } { q s } \\)

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For two points \((x_1,y_1)\) and \((x_2,y_2)\) on a line, the slope is calculated as the ratio of the change in \(y\) (rise) to the change in \(x\) (run).

Step2: Analyze the slopes of line segments related to the line

• Let's assume for a line, if we have two right - angled triangles formed by the line (like the ones in the coordinate - plane figure). The slope of a non - vertical line is constant. If \(\overline{PR}\) and \(\overline{QS}\) are line segments that represent the "rise - over - run" for the same line (using similar right - angled triangles formed with the line), then \(m_{\overline{PR}}=\frac{y_R - y_P}{x_R - x_P}\) and \(m_{\overline{QS}}=\frac{y_S - y_Q}{x_S - x_Q}\). Since the line has a constant slope, the slope of \(\overline{PR}\) is equal to the slope of \(\overline{QS}\).
• Also, the slope of the line can be represented as \(\frac{QR}{UR}\) (where \(QR\) is the vertical change and \(UR\) is the horizontal change in one of the right - angled triangles formed with the line).

Answer:

The slope of \(\overline{PR}\) is equal to the slope of \(\overline{QS}\), The slope of the line is equal to \(\frac{QR}{UR}\)