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.
statement
definition of slope
5 - 3 = 15 - 9
\\(\frac{2}{3}\\) = slope
\\(\frac{2}{3}\\) = \\(\frac{6}{9}\\)
the area of triangle a is greater than the area of triangle b.
reason
triangle a is similar to triangle b

Explanation:

Step1: Analyze slope equality

The slope between two points is $\frac{\Delta y}{\Delta x}$. For the slope between R and S, let's assume the $\Delta y = 5 - 3 = 2$ and $\Delta x = 3$ (from the triangle A), so slope is $\frac{2}{3}$? Wait, no, looking at the statements, we have $\frac{5}{3}=\frac{15}{9}$ (simplify $\frac{15}{9}=\frac{5}{3}$), so this shows the ratios of rise over run (slope) are equal. Also, $5 - 3 = 15 - 9$? Wait, $5 - 3 = 2$, $15 - 9 = 6$, no, that's not equal. Wait, the key is that similar triangles have proportional sides, so the slope (rise over run) should be equal. The statement "The area of triangle A is greater than the area of triangle B" – wait, no, the task is to match statements and reasons. Wait, the reason for slope between R and S equal to S and T: since triangles A and B are similar (Triangle A is similar to triangle B), their corresponding sides are proportional, so the slope (rise/run) is the same. Also, the definition of slope is rise over run. Let's match:

• "Definition of slope" should relate to slope formula.
• "$5 - 3 = 15 - 9$" – no, 2 ≠ 6, so wrong.
• "$\frac{5}{3} = \text{slope}$" – maybe for one triangle.
• "$\frac{5}{3} = \frac{15}{9}$" – simplifies to same, so this shows the slopes are equal (since $\frac{15}{9}=\frac{5}{3}$), so this is the proportion from similar triangles.
• "The area of triangle A is greater than the area of triangle B" – not about slope equality.
• "Triangle A is similar to triangle B" – reason why slopes are equal (similar triangles have proportional sides, so rise/run same).

Wait, the problem is to drag statements and reasons to show slope between R and S same as S and T. So:

Statement: The slope between R and S is the same as between S and T.

Reasons:

1. Definition of slope: slope is $\frac{\Delta y}{\Delta x}$.
2. $\frac{5}{3} = \frac{15}{9}$ (since $15/9 = 5/3$), so the rise over run (slope) is equal.
3. Triangle A is similar to triangle B (so corresponding sides are proportional, hence slope same).

But maybe the matching is:

• "Definition of slope" with slope formula.
• "$\frac{5}{3} = \frac{15}{9}$" as the proportion showing equal slopes.
• "Triangle A is similar to triangle B" as the reason for proportional sides.

But let's focus on the given options. The key is that $\frac{5}{3} = \frac{15}{9}$ (since 15÷3=5, 9÷3=3) shows the slopes are equal (rise over run), and triangle similarity implies proportional sides. Also, $5 - 3 = 15 - 9$ is incorrect (2≠6), so that's not a reason. The area statement is about area, not slope. So the correct matches would be:

• To show slope between R and S = S and T:

• Reason: Definition of slope (slope = $\frac{\Delta y}{\Delta x}$)
• Reason: $\frac{5}{3} = \frac{15}{9}$ (proportional slopes)
• Reason: Triangle A is similar to triangle B (so sides are proportional, hence slope same)

But maybe the specific task is to match the statement "The slope between R and S is the same as between S and T" with reasons. Let's assume the steps:

Step1: Recall slope definition

Slope is defined as $\frac{\text{change in } y}{\text{change in } x}$, so "Definition of slope" is a reason.

Step2: Show proportional slopes

$\frac{5}{3} = \frac{15}{9}$ because simplifying $\frac{15}{9}$ gives $\frac{5}{3}$, so this shows the rise over run (slope) is equal for both triangles.

Step3: Use triangle similarity

"Triangle A is similar to triangle B" implies their corresponding sides are proportional, so the slope (which is the ratio of sides) is the same.

Answer:

To show the slope between \( R \) and \( S \) equals the slope between \( S \) and \( T \):

• Statement: The slope between \( R \) and \( S \) is the same as between \( S \) and \( T \).
• Reasons:

1. Definition of slope: Slope is \( \frac{\Delta y}{\Delta x} \).
2. \( \boldsymbol{\frac{5}{3} = \frac{15}{9}} \): Simplifying \( \frac{15}{9} \) gives \( \frac{5}{3} \), so the rise-over-run (slope) is equal.
3. Triangle A is similar to triangle B: Similar triangles have proportional sides, so their slopes (side ratios) are equal.

(Note: The incorrect statement \( 5 - 3 = 15 - 9 \) is invalid, and "The area of triangle A is greater than triangle B" relates to area, not slope equality.)