Question

complete the parts below.
(a) find the rise, run, and slope given by triangle abc.
rise: 8 run: 10 slope: \\( \frac { 4 } { 5 } \\)
(b) find the rise, run, and slope given by triangle def.
rise: 3 run: 4 slope: \\( \frac { 3 } { 4 } \\)
(c) are the two slopes computed above equal? why or why not?
no. they are not equal because similar triangles can have different sizes.
no. they are not equal because the triangles are similar but not congruent.
yes. they are equal because the two triangles are congruent.
yes. they are equal because the two triangles are similar.

Explanation:

Step1: Recall the slope formula

The slope \(m\) of a line is given by \(m=\frac{\text{rise}}{\text{run}}\). For similar triangles, the ratios of corresponding sides are equal.

Step2: Analyze the slopes

For triangle \(ABC\), slope \(m_1 = \frac{8}{10}=\frac{4}{5}\). For triangle \(DEF\), slope \(m_2=\frac{3}{4}\). Wait, no! Wait, actually, if these are two triangles that are similar (assuming the problem is about similar - triangle - based slope). The slope is a ratio. For similar triangles, the ratio of rise to run (which is the slope) is the same. Because if \(\triangle ABC\sim\triangle DEF\), then \(\frac{\text{rise}_1}{\text{run}_1}=\frac{\text{rise}_2}{\text{run}_2}\) (by the property of similar triangles: corresponding sides are in proportion).

Answer:

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