Question

drag statements and reasons to each row to show why the slope of the line between r and s is the same as the slope between s and t, given that triangles a and b are similar. definition of slope $5 - 3 = 15 - 9$ the area of triangle a is greater than the area of triangle b. $\frac{5}{3} = $ slope $\frac{5}{3} = \frac{15}{9}$

Explanation:

Step1: Recall slope definition

Slope is defined as $\frac{\text{rise}}{\text{run}}$, i.e., $\frac{\text{change in } y}{\text{change in } x}$. For triangle A (between R and S), rise is 5, run is 3, so slope is $\frac{5}{3}$. For triangle B (between S and T), rise is 15, run is 9.

Step2: Use similarity of triangles

Since triangles A and B are similar, their corresponding sides are proportional. So $\frac{5}{3}=\frac{15}{9}$ (as corresponding rise over run ratios are equal due to similarity).

Step3: Conclude equal slopes

By definition of slope, slope between R and S is $\frac{5}{3}$, and slope between S and T is $\frac{15}{9}=\frac{5}{3}$, so they are equal.

The correct drag - and - drop steps (statements and reasons) would be:

• For the slope of RS: Use "Definition of slope" and $\frac{5}{3}=\text{slope}$.
• For the proportionality due to similar triangles: $\frac{5}{3}=\frac{15}{9}$.
• The statement $5 - 3=15 - 9$ is incorrect (not relevant), and "The area of triangle A is greater than the area of triangle B" is also incorrect (area isn't relevant for slope equality here).

Answer:

To show the slopes are equal:

1. For segment \(RS\): By "Definition of slope", the slope is \(\frac{5}{3}=\text{slope}\) (since rise = 5, run = 3).
2. For segments \(RS\) and \(ST\): Since triangles \(A\) and \(B\) are similar, \(\frac{5}{3}=\frac{15}{9}\) (corresponding sides of similar triangles are proportional, so rise - over - run ratios are equal).
3. Thus, the slope of \(RS\) (\(\frac{5}{3}\)) is equal to the slope of \(ST\) (\(\frac{15}{9}=\frac{5}{3}\)).

The incorrect statements are \(5 - 3 = 15 - 9\) (not a valid equality for slope reasoning) and "The area of triangle \(A\) is greater than the area of triangle \(B\)" (area does not determine slope equality).