Question

right triangles abc and a afe shown on the coordinate grid. which ratios are equivalent to the slope of ace? a) ab/bc and af/fe b) fe/ae and bc/ac c) bc/fa and ac/ae d) ab/ac and fe/ce

Explanation:

Step1: Recall the slope formula

The slope \(m\) of a line is given by \(m=\frac{\text{rise}}{\text{run}}\). For a right - triangle, if the right - triangle is used to find the slope of a line, the slope is the ratio of the length of the vertical side to the length of the horizontal side.
For right - triangle \(ABC\), the slope of the line (where \(AC\) is part of the line) is \(m = \frac{AB}{BC}\). For right - triangle \(AFE\), the slope of the line is \(m=\frac{FE}{DE}\) (since \(DE\) is the horizontal change and \(FE\) is the vertical change).
Since the two triangles are similar (by AA similarity, as the angles of the two right - triangles are equal because the lines are the same, so the corresponding angles are equal), the ratios of their corresponding sides are equal.

Step2: Analyze each option

• Option A:

The slope formula is \(\frac{\text{vertical}}{\text{horizontal}}\). For \(\frac{AB}{BC}\), \(AB\) is vertical and \(BC\) is horizontal in \(\triangle ABC\). For \(\frac{AF}{FE}\), \(AF\) is the hypotenuse (not the horizontal side) and \(FE\) is vertical. So \(\frac{AB}{BC}
eq\frac{AF}{FE}\).

• Option B:

For \(\frac{FE}{AE}\), \(AE\) is the hypotenuse (not the horizontal side) in \(\triangle AFE\), and for \(\frac{BC}{AC}\), \(AC\) is the hypotenuse (not the horizontal side) in \(\triangle ABC\). The slope is \(\frac{\text{vertical}}{\text{horizontal}}\), so \(\frac{FE}{AE}
eq\frac{BC}{AC}\).

• Option C:

For \(\frac{BC}{FA}\), \(BC\) is horizontal in \(\triangle ABC\) and \(FA\) is the hypotenuse in \(\triangle AFE\). For \(\frac{AC}{AE}\), \(AC\) and \(AE\) are hypotenuses. The slope formula is \(\frac{\text{vertical}}{\text{horizontal}}\), so \(\frac{BC}{FA}
eq\frac{AC}{AE}\).

• Option D:

In \(\triangle ABC\), the slope \(m = \frac{AB}{BC}\) (where \(AB\) is vertical and \(BC\) is horizontal). In \(\triangle DFE\) (a right - triangle similar to \(\triangle ABC\)), if we consider the line \(ACE\), the slope \(m=\frac{FE}{DE}\). Since \(\triangle ABC\sim\triangle DFE\) (by AA similarity, as the angles formed by the transversal with the parallel lines are equal), and \(DE = CE\) (by counting the grid units) and \(AB = FE\) (by counting the grid units in the vertical direction).

Answer:

D. AB/AC and FE/CE