Question

Lines a b c and d are shown below. Which lines are parallel? Which line are perpendicular? Explain and jutsify your answer using slopes

Answer 1

Lines A and C are parallel

Lines B and D are perpendicular

Explanation:

Given

See attachment for lines A to D

Required

Determine the parallel and perpendicular lines

To start with, we calculate the slope of each line.

Line A

The points on line A are:

`(x_1,y_1) = (3,-4)`

`(x_2,y_2) = (-1,2)`

Calculate the slope (m)

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (2 - (-4))/(-1 - 3)`

`m = (2 +4)/(-4)`

`m = (6)/(-4)`

`m_1 = -(3)/(2)`

Line B

`(x_1,y_1) = (-1,8)`

`(x_2,y_2) = (2,6)`

Calculate the slope (m)

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (6 - 8)/(2 -(-1))`

`m = (-2)/(2 +1)`

`m_2 = -(2)/(3)`

Line C

`y - 9 = -(3)/(2)(x + 8)`

Open bracket

`y - 9 = -(3)/(2)x -(3)/(2)* 8`

`y - 9 = -(3)/(2)x -3* 4`

`y - 9 = -(3)/(2)x -12`

Make y the subject

`y = -(3)/(2)x -12+9`

`y = -(3)/(2)x -3`

The slope intercept of an equation is:
`y = mx + b`

Where

`m = slope`

By comparison:

`m_3 = -(3)/(2)`

Line D

`12x - 8y = 40`

Subtract 12x from both sides

`12x-12x - 8y = -12x+40`

`- 8y = -12x+40`

Divide through by -8

`(- 8y)/(-8) = (-12x)/(-8)+(40)/(-8)`

`y= (12x)/(8)-(40)/(8)`

`y= (3)/(2)x-5`

The slope intercept of an equation is:
`y = mx + b`

Where

`m = slope`

By comparison:

`m_4 = (3)/(2)`

So, the slopes of the lines are:

`m_1 = -(3)/(2)`
`m_2 = -(2)/(3)`
`m_3 = -(3)/(2)`
`m_4 = (3)/(2)`

Lines with the same slope are parallel.

So:

Lines A and C with slope of
`-(3)/(2)` are parallel

Lines with the following relationship are perpendicular:

`m * M = -1`

Lines B and D are perpendicular because:

`-(2)/(3) * (3)/(2) = -1`

`-(6)/(6) = -1`

`-1 = -1`