Question

1. given a(-9, -12), b(-2, 2), c(x, 6), and d(-5, -2), find the value of x so that \\(\overline{ab} \parallel \overline{cd}\\).

Explanation:

Step1: Recall slope formula

The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Calculate slope of \(AB\)

For points \(A(-9,-12)\) and \(B(-2,2)\), the slope \(m_{AB}=\frac{2 - (-12)}{-2 - (-9)}=\frac{2 + 12}{-2 + 9}=\frac{14}{7} = 2\).

Step3: Calculate slope of \(CD\)

For points \(C(x,6)\) and \(D(-5,-2)\), the slope \(m_{CD}=\frac{-2 - 6}{-5 - x}=\frac{-8}{-5 - x}=\frac{8}{5 + x}\).

Step4: Set slopes equal (parallel lines have equal slopes)

Since \(AB\parallel CD\), \(m_{AB}=m_{CD}\). So, \(2=\frac{8}{5 + x}\).

Step5: Solve for \(x\)

Cross - multiply: \(2(5 + x)=8\).
Expand: \(10+2x = 8\).
Subtract 10 from both sides: \(2x=8 - 10=-2\).
Divide by 2: \(x=-1\).

Answer:

\(x = - 1\)