Question

which statements are true? select all that apply. the slope of \\( \overline { d b } \\) is equal to the slope of \\( \overline { g f } \\). the slope of \\( \overline { d b } \\) is equal to the slope of \\( \overline { g e } \\). the slope of \\( \overline { d b } \\) is equal to the slope of the line. the slope of the line is equal to \\( \frac { e f } { g f } \\). the slope of the line is equal to \\( \frac { b c } { d c } \\). the slope of the line is equal to \\( \frac { b c } { b d } \\).

Explanation:

Step1: Slope formula

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For two parallel lines, their slopes are equal. If \(\overline{DB}\), \(\overline{GE}\), \(\overline{GF}\) and the given line are parallel (implied by the problem context), then their slopes are equal.

Step2: Slope as a ratio

The slope of a line can also be represented as the ratio of the vertical change to the horizontal change. For a right - triangle formed by the line (e.g., in similar right - triangles \(\triangle BCD\) and \(\triangle EFG\)), the slope \(m=\frac{BC}{DC}=\frac{EF}{GF}\) (by the property of similar triangles, the ratios of corresponding sides are equal and slope is a ratio of vertical to horizontal change).

Answer:

The slope of \(\overline{DB}\) is equal to the slope of \(\overline{GF}\); The slope of \(\overline{DB}\) is equal to the slope of \(\overline{GE}\); The slope of \(\overline{DB}\) is equal to the slope of the line; The slope of the line is equal to \(\frac{EF}{GF}\); The slope of the line is equal to \(\frac{BC}{DC}\)