Question

what do the triangles tell you about the slope? complete each statement. the triangles are ?. the ratios of the corresponding side lengths are ?. the slope of each part of the line is ?.

Explanation:

Step1: Analyze Triangle Similarity

Triangles formed by the line and the grid (like \( \triangle AOE \), \( \triangle BFE \), \( \triangle CGF \), \( \triangle DGF \)) have corresponding angles equal (right angles and equal slope - related angles), so they are similar.

Step2: Calculate Corresponding Side Ratios

For \( \triangle AOE \): vertical change \( = 3 - 0 = 3 \)? Wait, no, looking at points: \( A(0,0) \), \( E(2,0) \), \( B(2,3) \)? Wait, no, from graph: \( A(0,0) \), \( E(2,0) \), \( B(2,3) \)? Wait, no, let's check coordinates. \( A(0,0) \), \( E(2,0) \), \( B(2,3) \): vertical leg length \( 3 - 0 = 3 \), horizontal leg length \( 2 - 0 = 2 \)? Wait, no, \( B \) is at (2,3)? Wait, the grid: from \( A(0,0) \) to \( E(2,0) \) (horizontal 2 units), \( E(2,0) \) to \( B(2,3) \) (vertical 3 units)? Wait, no, looking at the next triangle: \( B(2,3) \), \( F(4,3) \) (horizontal 2 units), \( F(4,3) \) to \( C(4,6) \) (vertical 3 units)? Wait, \( C(4,6) \), \( G(6,6) \) (horizontal 2 units), \( G(6,6) \) to \( D(6,9) \)? Wait, no, \( D \) is at (6,9)? Wait, the y - axis: \( D \) is at (6,9)? Wait, the graph shows \( D \) at (6,9)? Wait, no, the y - axis has 10 at the top. Wait, \( C \) is at (4,6), \( G \) at (6,6), \( D \) at (6,9). So for triangle \( AOE \): \( A(0,0) \), \( E(2,0) \), \( B(2,3) \)? Wait, no, \( B \) is at (2,3)? Then vertical change from \( A \) to \( B \) is 3, horizontal change is 2. For triangle \( BFC \): \( B(2,3) \), \( F(4,3) \) (horizontal 2), \( F(4,3) \) to \( C(4,6) \) (vertical 3). For triangle \( CGD \): \( C(4,6) \), \( G(6,6) \) (horizontal 2), \( G(6,6) \) to \( D(6,9) \) (vertical 3). So the vertical side lengths are 3, 3, 3 and horizontal are 2, 2, 2. So the ratio of vertical to horizontal for each is \( \frac{3}{2} \), and the triangles are similar (same shape, corresponding angles equal, sides proportional).

Step3: Determine Slope Consistency

Slope is \( \frac{\text{vertical change}}{\text{horizontal change}} \). For each segment: from \( A \) to \( B \): \( \frac{3 - 0}{2 - 0}=\frac{3}{2} \); from \( B \) to \( C \): \( \frac{6 - 3}{4 - 2}=\frac{3}{2} \); from \( C \) to \( D \): \( \frac{9 - 6}{6 - 4}=\frac{3}{2} \). So the slope is the same for each part.

Answer:

The triangles are \(\boldsymbol{\text{similar}}\).
The ratios of the corresponding side lengths are \(\boldsymbol{\text{equal}}\) (or specifically \(\boldsymbol{\frac{3}{2}}\) for vertical to horizontal, but "equal" in the sense of proportionality).
The slope of each part of the line is \(\boldsymbol{\frac{3}{2}}\) (or "the same" as they are equal).