Question

A (2,-4), B (3, 3) and C (-1,5) are the vertices of ΔABC . Find the equation of the altitude of the triangle through C.

Answer 1

x + 7y = 34

Explanation:

the altitude through C meets the opposite side AB at right angles

calculate the slope of AB using the slope formula

m =
`(y_(2)-y_(1) )/(x_(2)-x_(1) )`

with (x₁, y₁ ) = A (2, - 4 ) and (x₂, y₂ ) = B (3, 3 )

`m_(AB)` =
`(3-(-4))/(3-2)` =
`(3+4)/(1)` =
`(7)/(1)` = 7

given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/(7)` , then

y = -
`(1)/(7)` x + c ← partial equation of the altitude

to find c substitute C (- 1, 5 ) into the partial equation

5 =
`(1)/(7)` + c ⇒ c = 5 -
`(1)/(7)` =
`(35)/(7)` -
`(1)/(7)` =
`(34)/(7)`

y = -
`(1)/(7)` x +
`(34)/(7)` ( multiply through by 7 )

7y = - x + 34 ( add x to both sides )

x + 7y = 34 ← equation of altitude in standard form