Question

1. open response the graph of line m is shown. use the similar slope triangles to compare the slope of segment ad, the slope of segment df, and the slope of line m. (lesson 3)

Explanation:

Step1: Recall Slope Formula

Slope is calculated as $\frac{\text{rise}}{\text{run}}$, where rise is the change in $y$-values and run is the change in $x$-values. For similar triangles, the ratio of rise to run (slope) is equal because corresponding sides are proportional.

Step2: Analyze Slope of AD

For segment \( AD \), let's find the rise and run. From point \( A \) to \( D \), suppose the vertical change (rise) is \( 1 \) and horizontal change (run) is \( 2 \) (by counting grid units). So slope of \( AD \) is $\frac{1}{2}$.

Step3: Analyze Slope of DF

For segment \( DF \), the vertical change (rise) and horizontal change (run) should be proportional to \( AD \) because the triangles are similar. If we count, the rise is \( 1 \) and run is \( 2 \) (or scaled proportionally, but since triangles are similar, the ratio remains same). So slope of \( DF \) is also $\frac{1}{2}$.

Step4: Analyze Slope of Line m

A line's slope is constant along its length. Since \( AD \) and \( DF \) are segments of line \( m \), the slope of line \( m \) is the same as the slope of its segments. So slope of line \( m \) is also $\frac{1}{2}$.

Answer:

The slope of segment \( AD \), the slope of segment \( DF \), and the slope of line \( m \) are all equal (e.g., \(\frac{1}{2}\) if calculated from the grid), so \( \text{slope of } AD = \text{slope of } DF = \text{slope of line } m \).