To determine which ramp is steeper, Michael needs to calculate the slopes of both triangles, \( \triangle AGH \) and \( \triangle ABC \).
The slope \( m \) of a triangle can be calculated using the formula:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]
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Slope of Triangle \( \triangle AGH \):
- Let \( A \) be the bottom left vertex, \( G \) be the top vertex, and \( H \) be the bottom right vertex of triangle \( AGH \).
- Using the coordinates of points \( A(x_1, y_1) \), \( G(x_2, y_2) \), and \( H(x_3, y_3) \), the rise would be the change in the \( y \)-coordinates from \( A \) to \( G \), and the run would be the change in the \( x \)-coordinates from \( A \) to \( H \).
- It can be represented as: \[ m_{AGH} = \frac{y_G - y_A}{x_H - x_A} \]
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Slope of Triangle \( \triangle ABC \):
- Let \( A \) be the same bottom left vertex, \( B \) be the top vertex of triangle \( ABC \) (which might be at point \( G \)), and \( C \) be the bottom right vertex.
- The calculation for slope would be: \[ m_{ABC} = \frac{y_B - y_A}{x_C - x_A} \]
Once Michael calculates both slopes \( m_{AGH} \) and \( m_{ABC} \), he can compare the two values.
- If \( m_{ABC} > m_{AGH} \), then the ramp represented by triangle \( ABC \) is steeper than the ramp represented by triangle \( AGH \).
- Conversely, if \( m_{AGH} > m_{ABC} \), then the ramp represented by triangle \( AGH \) is the steeper option.
Please provide the specific coordinates of points A, G, H, B, and C, and I can help you calculate the exact slopes!