Question

Determine whether lines BT and MV are parallel, perpendicular, or neither. You must show all of your work to earn full credit.

B(1,-4), T(5,12), M(-8,3), V(-4,2)

Answer 1

the lines are perpendicular

to determine which case is true we require the slope m of the lines

Parallel lines have equal slopes

Perpendicular slopes are the negative inverse of each other

to calculate m use the gradient formula

m = ( y₂ - y₁ ) / ( x₂ - x₁ )

with (x₁, y₁ ) = B(1, - 4 ) and (x₂, y₂ ) = T(5, 12 )

`m_(BT)` =
`(12+4)/(5-1)` =
`(16)/(4)` = 4

repeat with

(x₁, y₁ ) = M(-8, 3 ) and (x₂, y₂ ) = V(-4, 2 )

`m_(MV)` =
`(2-3)/(-4+8)` = -
`(1)/(4)`

4 and -
`(1)/(4)` are negative inverses, hence

BT and MV are perpendicular to each other

Answer 2

BT ⊥ MV

Step-by-step explanation:

The direction vector BT is ...

... T - B = (5, 12) - (1, -4) = (4, 16)

The direction vector MV is ...

... V - M = (-4, 2) - (-8, 3) = (4, -1)

One is not a multiple of the other, so the lines are not parallel.

The dot-product of these direction vectors is

... (4, 16)•(4, -1) = 4·4 + 16·(-1) = 0

When the dot-product is zero, the vectors are perpendicular.