Question

A triangle has corners A, B, and C located at `(5 ,4 )`, `(2 ,3 )`, and `(5 ,9 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

End points of altitude through vertex C is `(5,9),(25/8, 27/8)`
Length of altitude is `5.9293`

Explanation:

Equation of line AB is
#(y-4)/(3-4)=(x-5)/(2-5)$ #-3y+12=-x+5# #y=(x/3)+(7/3)#

`3y-x=7color(white)(aaa)` Eqn (1)

Slope of AB is (1/3).
Slope of altitude is `-3`

Equation of altitude passing through corner C and perpendicular to side AB is,
`(y-9)=3*(x-5)`
`y-3x=-6 color (white)(aaa)` Eqn (2)

Solving Eqns (1), (2) we get the end point of altitude on AB.
`-9y+3x=-21`
`y-3x=-6`. Adding
`-8y=-27, y=27/8`
`(27/8)-3x=-6`
`-3x=-75/8, x=25/8`

Length of altitude`= sqrt((5-(25/8))^2+(9-(27/8))^2) = sqrt((15/8)^2+(45/8)^2)`
`=5.9293`