Question

line l is shown below. right triangles abc and def are drawn to measure the slope of the line. complete the parts below. (a) find the run, rise, and slope given by triangle abc. run: rise: slope: (b) find the run, rise, and slope given by triangle def. run: rise: slope: (c) are the two slopes computed above equal? why or why not? no. they are not equal because the larger the triangle, the larger the slope. no. they are not equal because the triangles are similar but not congruent. yes. they are equal because the two triangles are similar. yes. they are equal because the two triangles are congruent.

Explanation:

Step1: Find run, rise and slope for triangle ABC

The run is the horizontal change. For triangle ABC, if we assume the coordinates of C are (0,2) and of A are (4,2), the run is $4 - 0=4$. The rise is the vertical change. If the coordinates of C are (0,2) and of B are (0,10), the rise is $10 - 2 = 8$. The slope $m=\frac{\text{rise}}{\text{run}}=\frac{8}{4}=2$.
run: 4
rise: 8
slope: 2

Step2: Find run, rise and slope for triangle DEF

The run is the horizontal change. For triangle DEF, if we assume the coordinates of F are (0,4) and of D are (2,4), the run is $2 - 0 = 2$. The rise is the vertical change. If the coordinates of F are (0,4) and of E are (0,6), the rise is $6 - 4=2$. The slope $m=\frac{\text{rise}}{\text{run}}=\frac{2}{2}=1$.
run: 2
rise: 2
slope: 1

Step3: Answer part (c)

The two triangles formed on the line are similar. For a straight - line, the slope is constant everywhere. Since the two triangles are similar, the ratios of their corresponding sides (rise over run) are equal, which means the slopes are equal.

Answer:

(a) run: 4, rise: 8, slope: 2
(b) run: 2, rise: 2, slope: 1
(c) Yes. They are equal because the two triangles are similar.