Question

1. triangles pqr and rst are similar right triangles.

which proportion can be used to show that the slope of (overline{pr}) is equal to the slope of (overline{rt})?
a. (\frac{3 - 7}{-4 - (-7)}=\frac{-5 - 3}{2 - (-4)})
b. (\frac{3 - (-4)}{7 - (-7)}=\frac{-5 - 2}{3 - (-4)})
c. (\frac{-4 - (-7)}{3 - 7}=\frac{2 - (-4)}{-5 - 3})
d. (\frac{-4 - (-3)}{-7 - 7}=\frac{2 - (-5)}{-4 - 3})

Explanation:

Step1: Find coordinates of points

Let \(P(-7,7)\), \(R(-4,3)\), \(S(-4, - 4)\), \(T(2,-5)\).
The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
The slope of \(\overline{PR}\) with \(P(x_1,y_1)=(-7,7)\) and \(R(x_2,y_2)=(-4,3)\) is \(m_{PR}=\frac{3 - 7}{-4-(-7)}\).
The slope of \(\overline{RT}\) with \(R(x_1,y_1)=(-4,3)\) and \(T(x_2,y_2)=(2,-5)\) is \(m_{RT}=\frac{-5 - 3}{2-(-4)}\).

Step2: Check the proportion

Since the slopes are equal (\(m_{PR}=m_{RT}\)), the proportion \(\frac{3 - 7}{-4-(-7)}=\frac{-5 - 3}{2-(-4)}\) holds.

Answer:

A. \(\frac{3 - 7}{-4-(-7)}=\frac{-5 - 3}{2-(-4)}\)