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
the area of triangle a is greater than the area of triangle b
\frac{2}{3} = slope
\frac{2}{3} = \frac{10}{15}
reason
triangle a is similar to triangle b

Explanation:

Step1: Recall Slope Definition

Slope \( m \) is defined as \( m = \frac{\text{rise}}{\text{run}}=\frac{y_2 - y_1}{x_2 - x_1} \). For similar triangles, corresponding sides are proportional, so the ratio of rise to run (slope) should be equal.

Step2: Analyze Each Statement - Reason Pair

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

Reason: Definition of slope
Slope is rise over run. For the line, between any two points, slope is consistent (by definition) when triangles are similar (proportional sides).

• Statement: \( 5 - 3 = 15 - 9 \)

Simplify: \( 2 = 6 \)? No, incorrect.

• Statement: \( \frac{2}{3}=\text{slope} \)

If rise \( = 2 \), run \( = 3 \), slope is \( \frac{2}{3} \), but need to check proportionality.

• Statement: \( \frac{2}{3}=\frac{6}{9} \)

Simplify \( \frac{6}{9}=\frac{2}{3} \) (divide numerator/denominator by 3). This shows proportionality (similar triangles have proportional sides, so slope ratios are equal).

• Statement: Area of \( A \) > Area of \( B \)

Area depends on square of scale factor. If scale factor is \( \frac{1}{3} \) (from \( \frac{2}{6}=\frac{1}{3} \)), area ratio is \( \frac{1}{9} \), so \( A \) (smaller triangle) has less area. Incorrect.

Answer:

To show the slope between \( R \) and \( S \) is same as between \( S \) and \( T \) (given similar triangles):

• Statement: The slope between \( R \) and \( S \) is the same as the slope between \( S \) and \( T \)
• Reason: Definition of slope (slope is rise over run, and similar triangles have proportional rise/run, so slope is constant).

For proportionality (supporting similar triangles imply equal slope):

• Statement: \( \frac{2}{3}=\frac{6}{9} \)
• Reason: (Implied by similar triangles: corresponding sides are proportional, so \( \frac{\text{rise}_1}{\text{run}_1}=\frac{\text{rise}_2}{\text{run}_2} \), here \( \frac{2}{3}=\frac{6}{9} \) shows proportionality, hence equal slope.)

(Note: The correct pairing uses the definition of slope for the slope equality statement, and the proportionality \( \frac{2}{3}=\frac{6}{9} \) to show similar triangles lead to equal slope via proportional rise/run.)