Question

8.4a - use similar right triangles to develop an understanding that slope, m, given as the rate comparing the change in y - values to the change in x - values, (y2 - y1)/(x2 - x1), is the same for any two points (x1,y1) and (x2,y2) on the same line
#7
triangle abc and triangle xyz are similar right triangles.
which proportion can be used to show that the slope of ac is equal to the slope of xz?
a \\( \frac { - 4 - ( - 7 ) } { - 2 - ( - 4 ) } = \frac { 8 - 2 } { 6 - 2 } \\)
b \\( \frac { - 2 - ( - 4 ) } { - 4 - ( - 7 ) } = \frac { 2 - 6 } { 8 - 2 } \\)
c \\( \frac { - 2 - ( - 4 ) } { - 4 - ( - 7 ) } = \frac { 6 - 2 } { 8 - 2 } \\)
d \\( \frac { - 4 - ( - 2 ) } { - 4 - ( - 7 ) } = \frac { 6 - } { 8 - } \\)

Explanation:

Step1: Find the slope of \(AC\)

For points \(A(-7,-4)\) and \(C(-4,-2)\), the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) gives \(m_{AC}=\frac{-2-(-4)}{-4-(-7)}\).

Step2: Find the slope of \(XZ\)

For points \(X(2,2)\) and \(Z(8,6)\), the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\) gives \(m_{XZ}=\frac{6 - 2}{8 - 2}\).

Step3: Set up the proportion

Since \(m_{AC}=m_{XZ}\), the proportion is \(\frac{-2-(-4)}{-4-(-7)}=\frac{6 - 2}{8 - 2}\).

Answer:

C. \(\frac{-2-(-4)}{-4-(-7)}=\frac{6 - 2}{8 - 2}\)