Question

the triangles formed on the line are similar. which statements explain the relationship between the triangles and the slope? select all that apply.
options:

• $\frac{-4}{2} = -\frac{1}{2}$
• $\frac{-1}{2} = -\frac{2}{4}$
• $\frac{-2}{1} = -\frac{2}{4}$
• $\frac{-2}{1} = -\frac{4}{2}$
• $\frac{-2}{4} = -\frac{1}{2}$

Explanation:

Step1: Calculate slope of red triangle

Slope = $\frac{\text{rise}}{\text{run}}=\frac{-4}{2}=-2$

Step2: Calculate slope of green triangle

Slope = $\frac{\text{rise}}{\text{run}}=\frac{-2}{1}=-2$

Step3: Identify equal ratios

$\frac{-4}{2}=-2$ and $\frac{-2}{1}=-2$ are equal, so these statements apply.

Answer:

• $\boxed{-\frac{4}{2}=-\frac{1}{2}}$ (Note: Assuming the first option has a typo, likely $\boxed{-\frac{2}{4}=-\frac{1}{2}}$ and $\boxed{-\frac{4}{2}=-2}$ is incorrect; the correct slope from similar triangles is $\frac{\text{rise}}{\text{run}}=\frac{-2}{4}=-\frac{1}{2}$ or $\frac{-4}{2}=-2$ correction: wait, red triangle: run 2, rise -4 → slope $\frac{-4}{2}=-2$; green triangle: run 2, rise -1 → no, wait grid: green triangle run 2, rise -1? No, similar triangles have proportional sides: red triangle legs 4 (vertical down) and 2 (horizontal right) → slope $\frac{-4}{2}=-2$; green triangle legs 2 (vertical down) and 1 (horizontal right) → slope $\frac{-2}{1}=-2$? Wait the options: if an option is $\boxed{-\frac{4}{2}=-2}$ and $\boxed{-\frac{2}{1}=-2}$, those apply. But based on common similar triangle slope: slope = $\frac{\text{vertical change}}{\text{horizontal change}}$ for both triangles, so their ratios are equal.

(Note: Due to possible image option clarity, the key is similar triangles have equal slope ratios: $\frac{-4}{2}=\frac{-2}{1}=-2$ or $\frac{-2}{4}=\frac{-1}{2}$ if legs are reversed. Assuming the correct options are those where the fractions equal the slope, e.g., $\boxed{-\frac{4}{2}=-2}$ and $\boxed{-\frac{2}{1}=-2}$ if present.)