Question

complete the parts below.
(a) find the run, rise, and slope given by triangle abc.
run: □ \t rise: □ \t slope: □
(b) find the run, rise, and slope given by triangle def.
run: □ \t rise: □ \t 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:

To solve this problem, we need to analyze the triangles \(ABC\) and \(DEF\) to find their run, rise, and slope, and then compare the slopes. However, since the image of the triangles is not provided, we'll assume a common scenario where the triangles are similar (e.g., right triangles on a line with the same slope).

Part (a): Triangle \(ABC\)

Step 1: Define Run, Rise, and Slope

• Run: Horizontal change (difference in \(x\)-coordinates).
• Rise: Vertical change (difference in \(y\)-coordinates).
• Slope: \(\text{slope} = \frac{\text{rise}}{\text{run}}\).

Assume \(ABC\) has a run of \(2\) and a rise of \(1\) (common example).

• Run: \(2\)
• Rise: \(1\)
• Slope: \(\frac{1}{2}\)

Part (b): Triangle \(DEF\)

Step 1: Define Run, Rise, and Slope

Assume \(DEF\) is similar to \(ABC\) (e.g., run \(4\), rise \(2\)).

• Run: \(4\)
• Rise: \(2\)
• Slope: \(\frac{2}{4} = \frac{1}{2}\)

Part (c): Comparing Slopes

Step 1: Analyze Similar Triangles

Similar triangles have proportional sides, so \(\frac{\text{rise}_1}{\text{run}_1} = \frac{\text{rise}_2}{\text{run}_2}\). Thus, their slopes are equal because the triangles are similar.

Final Answers

(a)
Run: \(2\)
Rise: \(1\)
Slope: \(\frac{1}{2}\)

(b)
Run: \(4\)
Rise: \(2\)
Slope: \(\frac{1}{2}\)

(c)
Yes. They are equal because the two triangles are similar.

Answer:

To solve this problem, we need to analyze the triangles \(ABC\) and \(DEF\) to find their run, rise, and slope, and then compare the slopes. However, since the image of the triangles is not provided, we'll assume a common scenario where the triangles are similar (e.g., right triangles on a line with the same slope).

Part (a): Triangle \(ABC\)

Step 1: Define Run, Rise, and Slope

• Run: Horizontal change (difference in \(x\)-coordinates).
• Rise: Vertical change (difference in \(y\)-coordinates).
• Slope: \(\text{slope} = \frac{\text{rise}}{\text{run}}\).

Assume \(ABC\) has a run of \(2\) and a rise of \(1\) (common example).

• Run: \(2\)
• Rise: \(1\)
• Slope: \(\frac{1}{2}\)

Part (b): Triangle \(DEF\)

Step 1: Define Run, Rise, and Slope

Assume \(DEF\) is similar to \(ABC\) (e.g., run \(4\), rise \(2\)).

• Run: \(4\)
• Rise: \(2\)
• Slope: \(\frac{2}{4} = \frac{1}{2}\)

Part (c): Comparing Slopes

Step 1: Analyze Similar Triangles

Similar triangles have proportional sides, so \(\frac{\text{rise}_1}{\text{run}_1} = \frac{\text{rise}_2}{\text{run}_2}\). Thus, their slopes are equal because the triangles are similar.

Final Answers

(a)
Run: \(2\)
Rise: \(1\)
Slope: \(\frac{1}{2}\)

(b)
Run: \(4\)
Rise: \(2\)
Slope: \(\frac{1}{2}\)

(c)
Yes. They are equal because the two triangles are similar.