Question

geometry
4.5 equations of lines: slope - intercept &
standard form
warm up
determine whether \\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { c d } \\) would be parallel,
perpendicular, or neither.

1. \\( a ( - 4,8 ), b ( 4,10 ), c ( 1,1 ), d ( - 2,13 ) \\)
2. \\( a ( - 1, - 8 ), b ( 4, - 3 ), c ( - 3,1 ), d ( 5,9 ) \\)
3. \\( a ( - 5,3 ), b ( - 5,7 ), c ( 1,9 ), d ( - 10,9 ) \\)

definitions
equations of lines
most linear equations are written slope - intercept form or standard form.

Explanation:

Step1: Calculate slope of AB

The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For \(A(-4,8)\) and \(B(4,10)\), \(m_{AB}=\frac{10 - 8}{4-(-4)}=\frac{2}{8}=\frac{1}{4}\)

Step2: Calculate slope of CD

For \(C(1,1)\) and \(D(-2,13)\), \(m_{CD}=\frac{13 - 1}{-2 - 1}=\frac{12}{-3}=- 4\)

Step3: Check relationship

Since \(m_{AB}\times m_{CD}=\frac{1}{4}\times(-4)=-1\), the lines are perpendicular.

Step1: Calculate slope of AB

For \(A(-1,-8)\) and \(B(4,-3)\), \(m_{AB}=\frac{-3-(-8)}{4-(-1)}=\frac{5}{5} = 1\)

Step2: Calculate slope of CD

For \(C(-3,1)\) and \(D(5,9)\), \(m_{CD}=\frac{9 - 1}{5-(-3)}=\frac{8}{8}=1\)

Step3: Check relationship

Since \(m_{AB}=m_{CD} = 1\), the lines are parallel.

Step1: Analyze AB

For \(A(-5,3)\) and \(B(-5,7)\), \(x_1=x_2=-5\), so \(AB\) is a vertical line (\(x=-5\))

Step2: Analyze CD

For \(C(1,9)\) and \(D(-10,9)\), \(y_1 = y_2=9\), so \(CD\) is a horizontal line (\(y = 9\))

Step3: Check relationship

A vertical and a horizontal line are perpendicular.

Answer:

Perpendicular