Question

Michael is building a skateboard ramp. He cannot decide how long the ramp should be so he draws the blueprint below. Michael draws a vertical line in the big triangle ΔAGH to form the smaller triangle ΔABC. Michael determines the slope of both triangles to find out which ramp will be steeper.

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Answer 1

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:

\[
m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
\]

1. **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:
\[
m_{AGH} = \frac{y_G - y_A}{x_H - x_A}
\]

2. **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!