Question

A is the point (-2,0)

B is the point (0,4)
C is the point (5,-1)

Find an equation of the line that passes through C and is perpendicular to AB

Answer 1

y = 1/2x - 3.5

Explanation:

so lets start by finding the equation of the line AB

we know the y-intercept is 4 from the point 0,4 so we know it'll look something like this

y = mx + 4 and we can plug in the other point we know -> 0 = m(-2) + 4 -> 4 = m(-2) -> m = -2 so the full equation of the line is y = -2x + 4

ok now we can find the other line. we know that perpendicular lines have an opposire reciprocal slope so in this case -2 would turn into 1/2

y = 1/2x + b

now lets plug in point C

-1 = (1/2)(5) + b

-1 - (2.5) = b

-3.5 = b

so the equation is y = 1/2x - 3.5

Answer 2

`y=-\frac12 x + \frac32`

Explanation:

Equation of line AB:

Let A =
`(x_1,y_1)` = (-2, 0)

Let B =
`(x_2, y_2)` = (0, 4)

Use equation of slope to find slope m:

`m=(y_2-y_1)/(x_2-x_1)=(4 -0)/(0+2)=2`

Use point-slope form for linear equation:

`y-y_1=m(x-x_1)`

`\implies y-0=2(x+2)`

`\implies y =2x+4`

Equation of line passing through C

If the line that passes through C is perpendicular to AB, then their slopes will be opposite reciprocals of each other.

⇒
`m=(-1)/(2)=-\frac12`

Use point-slope form for linear equation and
`(x_1,y_1)` is point C:

`y-y_1=m(x-x_1)`

`\implies y+1=-\frac12 (x-5)`

`\implies y=-\frac12 x + \frac32`