Question

here are three ramps that make a smoothwhich statement about the slopes of the ramps is true?the slope of the first ramp is \\( \frac { 2 } { 3 } \\) the slope of the secondramp because the height of the first ramp is \\( \frac { 2 } { 3 } \\) thelength of the second ramp.the slope of the first ramp is the same as the slope of thesecond ramp because their triangles are similar.the slope of the first ramp is \\( \frac { 4 } { 9 } \\) the slope of the secondramp because the area of the first ramp triangle is \\( \frac { 4 } { 9 } \\) thearea of the second ramp triangle.the slope of the second ramp is 2 more than the slope ofthe first ramp because the difference between the basesof the ramps is 2.

Explanation:

Step1: Recall the slope formula

The slope formula is \(m = \frac{y}{x}\) (where \(y\) is the rise and \(x\) is the run). For similar triangles, if two right - angled triangles (representing the ramps) are similar, the ratio of their corresponding sides (rise and run) is the same.

Step2: Analyze the first option

Slope is \(\frac{\text{height}}{\text{length}}\). Just because the height of the first ramp is \(\frac{2}{3}\) the length of the second ramp does not relate the slope of the first ramp to the slope of the second ramp in the way described. The slope formula requires the ratio of height to length for the same ramp.

Step3: Analyze the second option

If the triangles (representing the ramps) are similar, then \(\frac{y_1}{x_1}=\frac{y_2}{x_2}\). Let the first ramp have height \(y_1\) and length \(x_1\), and the second ramp have height \(y_2\) and length \(x_2\). For similar triangles, the ratio of the height to the length (which is the slope \(m=\frac{y}{x}\)) is the same.

Step4: Analyze the third option

The area of a right - angled triangle is \(A=\frac{1}{2}xy\). If \(A_1=\frac{4}{9}A_2\), then \(\frac{1}{2}x_1y_1=\frac{4}{9}\times\frac{1}{2}x_2y_2\), or \(x_1y_1 = \frac{4}{9}x_2y_2\). But slope is \(\frac{y}{x}\), and this area relationship does not translate to a slope relationship as described.

Step5: Analyze the fourth option

Slope is a ratio (\(\frac{y}{x}\)), not a difference. Just because the difference in the bases (runs) is 2 does not mean the slope relationship is as described.

Answer:

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