Question

A triangle has corners A, B, and C located at `(1 ,4 )`, `(2 ,1 )`, and `(5 , 2 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

Endpoints of altitude are `(2,1) and (5,2)`
length of altitude is `sqrt10`

Explanation:

To calculate d the distance between two points, use the distance formula :
`d=sqrt((x_2−x_1)^2+(y_2−y_1)^2)`, where `(x_1,y_1)` and `(x_2,y_2)` are the coordinates of the two points.

Given `A(1,4), B(2,1), and C(5,2)`,
`AC=sqrt((5-1)^2+(2-4)^2)=sqrt20, => AC^2=20`
`AB=sqrt((2-1)^2+(1-4)^2)=sqrt10, => AB^2=10`
`BC=sqrt((5-2)^2+(2-1)^2)=sqrt10, => BC^2=10`
As `AC^2=AB^2+BC^2, Delta ABC` is a right triangle, right-angled at `B`.

Hence, `BC` is the altitude perpendicular to `AB` from `C`
`=>` endpoints of the altitude are `(2,1) and (5,2)`
length of the altitude `= BC = sqrt10`