Question

A triangle has corners A, B, and C located at `(4 ,7 )`, `(3 ,2 )`, and `(2 ,4 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

End points of altitude CD `(2,4) & (-7,97/26)`
Length of altitude CD =13.39#

Explanation:

Equation of AB is obtained using coordinates of A & B
`(y-7)/(2-7)=(x-4)/(3-4)`
`(y-7)/(-5)=(x-4)/(-1)`
`y-7=5(x-4)`
`y=5x-13`

Slope of AB `m1=5`
Slope of Altitude to line AB and passing through point C be m2.
`m2=-(1/m1)=-(1/5)`

Equal of Altitude passing through point C is
`(y-4)=-(1/5)*(x-2)`
`(-5y)+20=x-2`
`x+5y=22`

Solving equations of AB & Altitude through C gives base of altitude point D.
`y=5x-13`
`-5x+y=-13` Eqn (1)
`x+5y=22`
`5x+25y=110` Eqn (2)
Adding (1) & (2),
`26y=97`
`y=97/26`
`-5x+(97/26)=-13`
`-5x=-(97+338)/13=-(435/13)=-35`
`x=-7`
Coordinates of point D(-7,97/26)

Length of altitude CD is
`=sqrt((2-(-7))^2+(4-(97/26))^2)`
`sqrt(81+98.38)=13.39`