Question

in the figure below, $overleftrightarrow{pq}$ and $overleftrightarrow{rs}$ are parallel.

complete the following.
(a) find the slope of $overleftrightarrow{pq}$ and the slope of $overleftrightarrow{rs}$.
slope of $overleftrightarrow{pq}$: $\bigcirc -\frac{tq}{tp}$ $\bigcirc \frac{tp}{tq}$ $\bigcirc \frac{tq}{tp}$ $\bigcirc -\frac{tp}{tq}$
slope of $overleftrightarrow{rs}$: $\bigcirc \frac{tr}{ts}$ $\bigcirc \frac{ts}{tr}$ $\bigcirc -\frac{ts}{tr}$ $\bigcirc -\frac{tr}{ts}$
(b) it can be shown that $delta tpq sim delta trs$.
based on this, choose the ratio that is equal to $\frac{tp}{tq}$.
$\bigcirc \frac{ts}{tr}$ $\bigcirc \frac{tr}{rs}$ $\bigcirc \frac{rs}{ts}$ $\bigcirc \frac{tr}{ts}$
(c) using the results above, choose the correct statement below.
$\bigcirc$ slope of $overleftrightarrow{pq} \times$ slope of $overleftrightarrow{rs} = 1$
$\bigcirc$ slope of $overleftrightarrow{pq} =$ slope of $overleftrightarrow{rs}$
$\bigcirc$ slope of $overleftrightarrow{pq} \times$ slope of $overleftrightarrow{rs} = -1$
$\bigcirc$ slope of $overleftrightarrow{pq} = -$ slope of $overleftrightarrow{rs}$
(d) the result in part (c) is an example of the following rule for any two non-vertical parallel lines.
$\bigcirc$ the slopes of the two lines are reciprocals.
$\bigcirc$ the slopes of the two lines are negative reciprocals.
$\bigcirc$ the slopes of the two lines are the same.
$\bigcirc$ the slopes of the two lines are opposites.

Explanation:

Step1: Define slope formula

Slope = $\frac{\text{vertical change}}{\text{horizontal change}}$

Step2: Calculate slope of $\overleftrightarrow{PQ}$

Vertical change is $TP$, horizontal change is $TQ$. The line rises right, so slope is positive: $\frac{TP}{TQ}$

Step3: Calculate slope of $\overleftrightarrow{RS}$

Vertical change is $TR$, horizontal change is $TS$. The line rises right, so slope is positive: $\frac{TR}{TS}$

Step4: Use similar triangles $\Delta TPQ \sim \Delta TRS$

Corresponding sides are proportional: $\frac{TP}{TQ} = \frac{TR}{TS}$

Step5: Relate slopes of parallel lines

From similar triangles, their slopes are equal. This matches the rule for parallel lines.

Answer:

(a)
slope of $\overleftrightarrow{PQ}$: $\frac{TP}{TQ}$
slope of $\overleftrightarrow{RS}$: $\frac{TR}{TS}$

(b)
$\frac{TR}{TS}$

(c)
slope of $\overleftrightarrow{PQ}$ = slope of $\overleftrightarrow{RS}$

(d)
The slopes of the two lines are the same.