Question

triangle def and fgh are shown on the coordinate plane below. select the options that correctly complete the sentence. the slope of line segment df is equal to the slope of line segment fh because the ratio of select answer to the ratio of select answer.

Explanation:

Step1: Recall Slope Formula

The slope of a line segment between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\), which is the ratio of the vertical change (rise) to the horizontal change (run).

Step2: Analyze Segments DF and FH

For line segment \(DF\) (in triangle \(DEF\)) and line segment \(FH\) (in triangle \(FGH\)), since they are part of the same line (collinear, as their slopes are equal), the slope of \(DF\) is \(\frac{\text{rise of } DF}{\text{run of } DF}\) and the slope of \(FH\) is \(\frac{\text{rise of } FH}{\text{run of } FH}\). For the slopes to be equal, the ratio of (rise of \(DF\)) to (run of \(DF\)) must equal the ratio of (rise of \(FH\)) to (run of \(FH\)). So the first "Select answer" should be related to the rise and run of \(DF\), and the second to the rise and run of \(FH\) (e.g., if options for first are "rise of \(DF\) to run of \(DF\)" and second "rise of \(FH\) to run of \(FH\)").

Answer:

(Assuming options: First dropdown: "rise of \(DF\) to run of \(DF\)", Second dropdown: "rise of \(FH\) to run of \(FH\)")
First: rise of \(DF\) to run of \(DF\)
Second: rise of \(FH\) to run of \(FH\)