Question

d a certain number of words per minute to complete his essay. the if words he typed is modeled using a straight line on a coordinate. the length of $overline{qr}$ is 150. the length of $overline{pr}$ is 3. the slope of $overline{pq}$ is the height - to - width ratio of $\triangle pqr$. enter the slope of $overline{pq}$: 50. points $q$, $s$, and $t$ form $\triangle qst$. use the triangle to complete each statement. enter the length of $overline{st}$: enter the length of $overline{qt}$: the slope of $overline{qs}$ is the height - to - width ratio of $\triangle qst$. enter the slope of $overline{qs}$:

Explanation:

Step1: Determine the slope - concept

The slope of a line is the ratio of the vertical change (rise) to the horizontal change (run). Given that the slope of $\overline{PQ}$ is 50 (from the text), and assuming the line has a constant slope (since it's a straight - line model).

Step2: Analyze the grid for $\overline{ST}$

Looking at the grid, if we assume the horizontal distance (run) between points $S$ and $T$ is 2 units (by counting the grid squares on the x - axis). Since the slope of the line is 50, and slope $m=\frac{\text{rise}}{\text{run}}$. Let the length of $\overline{ST}$ (vertical distance or rise) be $y$ and run be $x = 2$. We know $m = 50=\frac{y}{x}$. Substituting $x = 2$ into the slope formula $y=m\times x$. So $y=50\times2 = 100$.

Step3: Analyze the grid for $\overline{QT}$

The horizontal distance (run) between $Q$ and $T$ is 5 units (by counting the grid squares on the x - axis). Using the slope formula $m=\frac{\text{rise}}{\text{run}}$ with $m = 50$ and $x = 5$. Then the vertical distance (rise) which is the length of the vertical part of $\overline{QT}$ is $y=m\times x=50\times5 = 250$. And the horizontal distance between $Q$ and $T$ is 5. Using the Pythagorean theorem $d=\sqrt{(250)^{2}+5^{2}}=\sqrt{62500 + 25}=\sqrt{62525}\approx250.05$. But if we consider the right - triangle formed by the vertical and horizontal displacements and we are just looking for the sum of the vertical and horizontal displacements in a non - Pythagorean sense (counting grid units along the path of the line), the length of $\overline{QT}$ is 250 (vertical displacement) as the horizontal displacement does not contribute to the length along the line in the context of the slope - based calculation.

Step4: Calculate the slope of $\overline{QS}$

The vertical change (rise) between $Q$ and $S$ is $400 - 200=200$, and the horizontal change (run) is $8 - 4 = 4$. The slope of $\overline{QS}$ is $\frac{200}{4}=50$.

Answer:

Length of $\overline{ST}$: 100
Length of $\overline{QT}$: 250
Slope of $\overline{QS}$: 50