Question

triangle def and triangle fgh are similar right triangles. the coordinates of all vertices are integers.

choose the correct answer from each drop down menu to complete the statement.

the slope of line df is choose... the slope of line fh, because the ratio of change in y - values to the change in x - values is choose...

Explanation:

Step1: Calculate the slope of line DF

For line DF, the change in \(y\)-values (\(\Delta y\)) is \(4 - 7=- 3\), and the change in \(x\)-values (\(\Delta x\)) is \(-2-(-7) = 5\). The slope \(m_{DF}=\frac{\Delta y}{\Delta x}=\frac{-3}{5}\)

Step2: Calculate the slope of line FH

For line FH, the change in \(y\)-values (\(\Delta y\)) is \(1 - 4=-3\), and the change in \(x\)-values (\(\Delta x\)) is \(8 - 3=5\). The slope \(m_{FH}=\frac{\Delta y}{\Delta x}=\frac{-3}{5}\)

Since the two triangles are similar, the ratios of the corresponding side - lengths (which are related to the changes in \(y\) and \(x\) values for the slope calculation) are equal.

Answer:

The slope of line DF is equal to the slope of line FH, because the ratio of change in y - values to the change in x - values is the same.