Question

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

Explanation:

Part (a)

Step1: Determine rise for ABC

Count vertical grid units from B to A: 4

Step2: Determine run for ABC

Count horizontal grid units from B to C: 1

Step3: Calculate slope for ABC

Slope = $\frac{\text{rise}}{\text{run}} = \frac{4}{1} = 4$

Part (b)

Step1: Determine rise for DEF

Count vertical grid units from E to D: 8

Step2: Determine run for DEF

Count horizontal grid units from E to F: 2

Step3: Calculate slope for DEF

Slope = $\frac{\text{rise}}{\text{run}} = \frac{8}{2} = 4$

Part (c)

Step1: Compare slopes

Slopes from (a) and (b) are both 4.

Step2: Identify triangle relationship

Triangles ABC and DEF are similar (right triangles with same angle at line l), so their rise-run ratios (slopes) are equal.

Step3: Select correct option

Choose the option stating equality due to similarity.

Answer:

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