Question

line l is shown below. right triangles abc and def are drawn to measure the slope of the line. 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 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:

Step1: Find rise, run and slope for triangle \(ABC\)

For triangle \(ABC\), the vertical change (rise) is \(4\) units and the horizontal change (run) is \(1\) unit. The slope \(m=\frac{\text{rise}}{\text{run}}\), so \(m = \frac{4}{1}=4\)

Step2: Find rise, run and slope for triangle \(DEF\)

For triangle \(DEF\), the vertical change (rise) is \(8\) units and the horizontal change (run) is \(2\) units. The slope \(m=\frac{\text{rise}}{\text{run}}\), so \(m=\frac{8}{2} = 4\)

Step3: Compare the slopes

Since the two triangles \(ABC\) and \(DEF\) are similar (by AA similarity, as the angles of the right - angled triangles are equal because the line \(l\) makes the same angles with the horizontal and vertical directions for both triangles). For similar triangles, the ratios of their corresponding sides (rise and run) are equal. So the slopes \(\frac{\text{rise}_1}{\text{run}_1}=\frac{\text{rise}_2}{\text{run}_2}\)

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.