Question

which statements are true? select all that apply.
the slope of \\(\overline{ab}\\) is equal to the slope of \\(\overline{bd}\\).
the slope of \\(\overline{ab}\\) is equal to the slope of \\(\overline{ac}\\).
the slope of \\(\overline{ab}\\) is equal to the slope of the line.
the slope of the line is equal to \\(\frac{ad}{bd}\\).
the slope of the line is equal to \\(\frac{ae}{ac}\\).
the slope of the line is equal to \\(\frac{ae}{ce}\\).

Explanation:

1. Slope of Collinear Segments: Segments on the same line (collinear) have equal slopes. $\overline{AB}$, $\overline{BD}$, $\overline{AC}$ are parts of the same line, so their slopes are equal. The slope of $\overline{AB}$ is also equal to the slope of the entire line.
2. Slope as Rise/Run: For a line, slope is $\frac{\text{rise}}{\text{run}}$. In right triangles formed (e.g., $\triangle ABD$, $\triangle AEC$), the ratio of vertical change to horizontal change (rise/run) equals the slope.

• $\frac{AD}{BD}$: $AD$ (vertical) and $BD$ (horizontal) form a right triangle with the line, so slope = $\frac{AD}{BD}$.
• $\frac{AE}{CE}$: $AE$ (vertical) and $CE$ (horizontal) form a right triangle with the line, so slope = $\frac{AE}{CE}$.
• $\frac{AE}{AC}$ is incorrect because $AC$ is not horizontal (it’s the hypotenuse, not the run), so this ratio does not represent slope. However, in the given options, the checked statements align with collinearity and correct rise/run ratios for the other two.

Answer:

• The slope of $\overline{AB}$ is equal to the slope of $\overline{BD}$.
• The slope of $\overline{AB}$ is equal to the slope of $\overline{AC}$.
• The slope of $\overline{AB}$ is equal to the slope of the line.
• The slope of the line is equal to $\frac{AD}{BD}$.
• The slope of the line is equal to $\frac{AE}{CE}$.

(Note: The statement "The slope of the line is equal to $\frac{AE}{AC}$" is incorrect, but the other five checked statements are true based on collinearity and slope definition.)