Question

1. which statement about the slopes of the ramps is true?

a. the slope of the first ramp is \\( \frac { 2 } { 3 } \\) the slope of the second ramp because the height of the first ramp is \\( \frac { 2 } { 3 } \\) the length of the second ramp.
b. the slope of the first ramp is the same as the slope of the second ramp because their triangles are similar.
c. the slope of the first ramp is \\( \frac { 4 } { 9 } \\) the slope of the second ramp because the area of the first ramp triangle is \\( \frac { 4 } { 9 } \\) the area of the second ramp triangle.
d. the slope of the second ramp is 2 more than the slope of the first ramp because the difference between the bases of the ramps is 2.

Explanation:

Step1: Recall the property of slopes of similar triangles

If two triangles (formed by the ramps) are similar, the ratio of their corresponding sides (which gives the slope) is the same.

Step2: Analyze each option

• Option A: Slope is not determined by height of the first ramp being \(\frac{2}{3}\) of the second. Slope is \(\frac{\text{rise}}{\text{run}}\).
• Option B: If the triangles (formed by the ramps) are similar, \(\text{Slope}=\frac{\text{vertical change}}{\text{horizontal change}}\) and for similar triangles \(\frac{\text{vertical change}_1}{\text{horizontal change}_1}=\frac{\text{vertical change}_2}{\text{horizontal change}_2}\), so slopes are same.
• Option C: Slope is not related to the area of the triangles in the way described. Slope formula has no relation to area (\(A = \frac{1}{2}\times base\times height\)) for slope determination.
• Option D: Slope is \(\frac{\text{rise}}{\text{run}}\), difference in bases does not directly give the difference in slopes in the described manner.

Answer:

B. The slope of the first ramp is the same as the slope of the second ramp because their triangles are similar.