Question

1. the question has two parts

part a: which ratios are equivalent to the slope of the line that passes through
point, a, e, and c?

Explanation:

To solve this, we analyze the slope of the line through \( A \), \( E \), and \( C \). The slope formula is \( \text{slope} = \frac{\text{change in } y}{\text{change in } x} \).

Step 1: Analyze \( \frac{AB}{BC} \)

• \( AB \) is the vertical change (rise) from \( A \) to \( B \).
• \( BC \) is the horizontal change (run) from \( B \) to \( C \).
• The slope of \( AC \) is \( \frac{AB}{BC} \) (since slope = rise/run).

Step 2: Analyze \( \frac{AD}{DE} \)

• \( AD \) is the vertical change (rise) from \( A \) to \( D \).
• \( DE \) is the horizontal change (run) from \( D \) to \( E \).
• The line \( DE \) is parallel to \( BC \) (both horizontal), and \( AD \) is parallel to \( AB \) (both vertical). By similar triangles (or proportionality), \( \frac{AD}{DE} = \frac{AB}{BC} \), so this ratio is also equal to the slope of \( AC \).

Thus, the ratios \( \boldsymbol{\frac{AB}{BC}} \) and \( \boldsymbol{\frac{AD}{DE}} \) are equivalent to the slope of the line through \( A \), \( E \), and \( C \).

Answer:

To solve this, we analyze the slope of the line through \( A \), \( E \), and \( C \). The slope formula is \( \text{slope} = \frac{\text{change in } y}{\text{change in } x} \).

Step 1: Analyze \( \frac{AB}{BC} \)

• \( AB \) is the vertical change (rise) from \( A \) to \( B \).
• \( BC \) is the horizontal change (run) from \( B \) to \( C \).
• The slope of \( AC \) is \( \frac{AB}{BC} \) (since slope = rise/run).

Step 2: Analyze \( \frac{AD}{DE} \)

• \( AD \) is the vertical change (rise) from \( A \) to \( D \).
• \( DE \) is the horizontal change (run) from \( D \) to \( E \).
• The line \( DE \) is parallel to \( BC \) (both horizontal), and \( AD \) is parallel to \( AB \) (both vertical). By similar triangles (or proportionality), \( \frac{AD}{DE} = \frac{AB}{BC} \), so this ratio is also equal to the slope of \( AC \).

Thus, the ratios \( \boldsymbol{\frac{AB}{BC}} \) and \( \boldsymbol{\frac{AD}{DE}} \) are equivalent to the slope of the line through \( A \), \( E \), and \( C \).