https://file.aiquickdraw.com/mathbot/file/17260051529689z69m2xr.jpg

Let's work through the questions based on the slope triangles provided in the graph.

### a. Calculate the slope using slope triangles A and B. What do you notice?
- **Slope Triangle A**:
  - Horizontal length (\(\Delta x\)): 4 units
  - Vertical length (\(\Delta y\)): 6 units
  - Slope of Triangle A = \(\frac{\Delta y}{\Delta x} = \frac{6}{4} = 1.5\)

- **Slope Triangle B**:
  - Horizontal length (\(\Delta x\)): 2 units
  - Vertical length (\(\Delta y\)): 3 units
  - Slope of Triangle B = \(\frac{\Delta y}{\Delta x} = \frac{3}{2} = 1.5\)

  **Observation**: The slopes of both triangles A and B are the same, which indicates that the line is linear and has a constant slope, regardless of the triangle size.

### b. What is the vertical distance (\(\Delta y\)) of slope triangle C? Explain your reasoning.
- **Slope Triangle C**:
  - Horizontal length (\(\Delta x\)): 1 unit
  - The slope is consistent, so using the formula for slope (\(m = 1.5\)):
  \[
  \Delta y = 1.5 \times 1 = 1.5
  \]
  Therefore, the vertical distance (\(\Delta y\)) of slope triangle C is 1.5 units. This maintains the constant slope of the line.

### c. Draw a slope triangle on the line with a horizontal length (\(\Delta x\)) of 1 unit. What is the vertical length (\(\Delta y\)) of this new triangle? How is this related to slope triangles A, B, and C?
- If we draw a new slope triangle with \(\Delta x = 1\) unit:
  - The slope is still 1.5, so:
  \[
  \Delta y = 1.5 \times 1 = 1.5
  \]
  The vertical distance is 1.5 units, consistent with the slope of triangles A, B, and C. This shows that the slope remains constant regardless of the size of the triangle.

### d. What happens to the slope when the slope triangles are different sizes?
- When the slope triangles are different sizes, the slope remains the same because the line is linear. The ratio of \(\Delta y\) to \(\Delta x\) does not change as long as the points lie on the same straight line, which indicates a constant slope.

Would you like a more detailed explanation of any part of this? Here are 5 related questions to think about:

1. How can we prove the slope formula \(\frac{\Delta y}{\Delta x}\) always works for straight lines?
2. Can you find an example where the slope of two triangles is not the same? What would that imply about the line?
3. What is the relationship between slope and the angle of the line with the x-axis?
4. How can you use this concept of slope to find the equation of the line?
5. Why is the slope for a vertical line undefined?

**Tip**: Always remember that slope measures the "steepness" of a line. If the slope is the same for different triangles, the line is straight and consistent.

Use the graphs below with slope triangles A, B, and C to answer the following questions: a. Calculate the slope using slope triangles A and B. What do you notice? b. What is the vertical distance (Δy) of slope triangle C? Explain your reasoning. c. Draw a slope triangle on the line with a horizontal length (Δx) of 1 unit. What is the vertical length (Δy) of this new triangle? How is this related to slope triangles A, B, and C? d. What happens to the slope when the slope triangles are different sizes?

Learn how to calculate the slope of a line using slope triangles A, B, and C. This problem covers key concepts in linear functions, including the slope formula and how similar triangles maintain a constant slope on a linear graph. Ideal for students in Grades 8-10.

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