Question

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?
yes. they are equal because the two triangles are congruent.
yes. they are equal because the two triangles are similar.
no. they are not equal because the larger the triangle, the larger the slope.
no. they are not equal because the larger the triangle, the smaller the slope.

Explanation:

Step1: Determine run and rise for triangle ABC

Count horizontal units from A to C for run and vertical units from A to B for rise.
From the graph, for triangle ABC, the run is the horizontal distance from A to C. If we assume A is at (0, 2) and C is at (4, 2), the run = 4 - 0=4. The rise is the vertical distance from A to B. If B is at (0, 6), the rise = 6 - 2 = 4. The slope formula is $m=\frac{\text{rise}}{\text{run}}$, so slope of triangle ABC $m_{ABC}=\frac{4}{4}=1$.

Step2: Determine run and rise for triangle DEF

Count horizontal units from D to F for run and vertical units from D to E for rise.
If D is at (12, 10) and F is at (16, 10), the run = 16 - 12 = 4. If E is at (12, 14), the rise = 14 - 10 = 4. The slope of triangle DEF $m_{DEF}=\frac{4}{4}=1$.

Step3: Compare the slopes

The two triangles are similar (not congruent as they have different side - lengths but same shape). For similar right - angled triangles formed by a line, the ratios of rise to run (slopes) are equal.

Answer:

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