Question

Are AB and CD parallel, perpendicular, or neither?*
1. A(-4, 8), B(4, 10), C(1, 1), D(-2, 13)

Answer 1

Perpendicular

Explanation:

Given

`A = (-4, 8)`

`B = (4, 10)`

`C = (1, 1)`

`D = (-2, 13)`

Required

Is AB and CD parallel?

First, we need to calculate the slope (m) of AB and CD

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

For AB:

`A (x_1,y_1)= (-4, 8)`

`B(x_2,y_2) = (4, 10)`

So:

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

`m = (10 - 8)/(4 - (-4))`

`m = (10 - 8)/(4 +4)`

`m = (2)/(8)`

`m = (1)/(4)`

For CD:

`C(x_1,y_1) = (1, 1)`

`D(x_2,y_2) = (-2, 13)`

So:

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

`m = (13 - 1)/(-2 - 1)`

`m = (12)/(-3)`

`m = -4`

For Lines to be parallel, the slope must be equal:

i.e.
`m_1 = m_2`

This condition is not true because:

`(1)/(4) \\eq -4`

For Lines to be perpendicular, the slope must be:

`m_1 = -(1)/(m_2)`

This implies:

`(1)/(4) = -(1)/(-4)`

`(1)/(4) = (-1)/(-4)`

`(1)/(4) = (1)/(4)`

Hence, the lines are perpendicular