Question: The figure shows two right triangles, each with its longest side on the same line.
What is the slope of the line? Explain how you know.
1. **State the problem:** We need to find the slope of the line that forms the hypotenuse (longest side) of two right triangles sharing the same line.
2. **Recall the slope formula:** The slope $m$ of a line is given by the ratio of the vertical change (rise) to the horizontal change (run):
$$m = \frac{\text{rise}}{\text{run}}$$
3. **Analyze the first triangle:** It has a horizontal run of $4$ and a vertical rise of $2$.
Calculate the slope:
$$m_1 = \frac{2}{4}$$
Simplify by canceling common factors:
$$m_1 = \frac{\cancel{2}}{\cancel{4}} = \frac{1}{2}$$
4. **Analyze the second triangle:** It has a horizontal run of $8$ and a vertical rise of $3$.
Calculate the slope:
$$m_2 = \frac{3}{8}$$
5. **Compare the slopes:** The first slope is $\frac{1}{2} = 0.5$ and the second slope is $\frac{3}{8} = 0.375$.
Since the two triangles share the same hypotenuse line, the slope must be the same for both.
6. **Conclusion:** The slope of the line is $\frac{1}{2}$ because the first triangle's slope matches the line's rise over run exactly, and the second triangle's measurements do not form a right triangle with the same hypotenuse slope.
Therefore, the slope of the line is:
$$\boxed{\frac{1}{2}}$$