Question

A triangle has corners A, B, and C located at `(4 ,7 )`, `(9 ,5 )`, and `(6 ,2 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

Endpoints `(6, 2)` and `(216/29,163/29)`
Length `l=1/29sqrt(12789)=3.8996`

Explanation:

One of the endpoints is point `C(6, 2)`. To solve for the other endpoint, we need to know the intersection of the lines containing points A and B, and line containing the altitude going thru C.

line AB equation by the two-point form

`y-y_a=(y_b-y_a)/(x_b-x_a)(x-x_a)`

Given `A(4, 7)` and `B(9, 5)` and `C(6, 2)`

`y-y_a=(y_b-y_a)/(x_b-x_a)(x-x_a)`

`y-7=(5-7)/(9-4)(x-4)`

`y-7=-2/5(x-4)`

`5y-35=-2x+8`

`2x+5y=43" "`first equation

Solve for the equation containing the altitude thru C using point-slope form

slope`m=-(1/(-2/5))=5/2` and point `C(6, 2)`

`y-y_c=m(x-x_c)`

`y-2=5/2(x-6)`

`2y-4=5x-30`

`5x-2y=26" "`second equation

Solve for the other endpoint using first and second equations

`2x+5y=43" "`first equation
`5x-2y=26" "`second equation

Simultaneous solution results to:

`x=216/29` and `y=163/29`

Solve for the length of the altitude:from `(216/29,163/29)` to point `C(6,2)`

`l=sqrt((6-216/29)^2+(2-163/29)^2)`
`l=sqrt((-42/29)^2+(-105/29)^2)`

`l=1/29sqrt(12789)`

`l=3.8996`

God bless...I hope the explanation is useful..