Question

A triangle has corners A, B, and C located at `(2 ,3 )`, `(5 ,8 )`, and `(3 , 2 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

Length of altitude passing through point C is `5.831`

Explanation:

Eqn of line AB is
`(y-y_1) / (y_2-y_1) = (x-x_1)/(x_2-x_1)`
`(y - 3)/(8-3) = (x-2)/(5-2)`
`3y-3 = 5x-10`
`5x-3y = 7`. Eqn (1)

Slope of line AB `=(y_2-y_1) / (x_2-x_1) = (2-3)/(3-2) = -1`
Slope of altitude passing through point C `= -(1/-1) = 1`
Eqn of Altitude passing through point C is
`(y-y_1) = m (x -x_1)`
`y - 2 = 1 * (x - 3)`
`x - y = 1` Eqn (2)

Solving Eqns (1) , (2) we get coordinates of altitude base passing through C.
`2y = 2, y = 1, x = 2`

Length of Altitude passing through C is
`sqrt( ((3+2)^2+(2+1)^2) = sqrt34 = 5.831`