Question

(a) find the rise, run, and slope given by triangle abc.
rise: 10 run: 8 slope: \\(\frac{5}{4}\\)
(b) find the rise, run, and slope given by triangle def.
rise: 5 run: 4 slope: \\(\frac{5}{4}\\)
(c) are the two slopes computed above equal? why or why not?
\\(\circ\\) no. they are not equal because the larger the triangle, the larger the slope.
\\(\circ\\) no. they are not equal because the triangles are similar but not congruent.
\\(\circ\\) yes. they are equal because the two triangles are similar.
\\(\circ\\) yes. they are equal because the two triangles are congruent.

Explanation:

Step1: Analyze Slope Values

From (a), slope of \( ABC \) is \( \frac{5}{4} \) (since \( \frac{10}{8}=\frac{5}{4} \) after simplifying by dividing numerator and denominator by 2). From (b), slope of \( DEF \) is \( \frac{5}{4} \). So the slopes are equal.

Step2: Relate to Triangle Similarity

Similar triangles have corresponding sides in proportion. For \( ABC \) and \( DEF \), the rise (vertical side) and run (horizontal side) of \( ABC \) are \( 10 \) and \( 8 \), and for \( DEF \) are \( 5 \) and \( 4 \). The ratio of rise to run (slope) is the same, and the triangles are similar (since \( \frac{10}{5}=\frac{8}{4}=2 \), so sides are in proportion). Slope depends on the ratio of rise to run, so similar triangles (same shape, proportional sides) have the same slope. The triangles are not congruent (since sides are not equal, just proportional), but similarity ensures the slope (ratio) is equal.

Answer:

Yes. They are equal because the two triangles are similar.