Question

A triangle has corners A, B, and C located at `(5 ,2 )`, `(2 ,5 )`, and `(3 ,7 )`, respectively. What are the endpoints and length of the altitude going through corner C?

Answer 1

Endpoints of altitude `CD` are `(3,7) and (3/2,11/2)`
Length of altitude `CD` is `2.12` unit

Explanation:

`A(5,2) , B(2,5) , C(3,7)`

Let `CD` be the altitude going through `C` touches `D` on line

`AB`. `C` and `D` are the endpoints of altitude `CD; CD` is

perpendicular on `AB`. Slope of `AB= m_1= (y_2-y_1)/(x_2-x_1)`

`=(5-2)/(2-5) = 3/(-3)= -1 :.` Slope of `CD=m_2= -1/m_1= 1`

Equation of line `AB` is `y - y_1 = m_1(x-x_1)`or

`y-2 = -1(x-5) or x +y = 7 ; (1)`

Equation of line `CD` is `y - y_3 = m_2(x-x_3)` or

`y- 7 = 1(x-3) or x -y = -4 (2)` Solving equation (1) and

(2) we get the co-ordinates of `D`

Adding equation (1) and (2) we get `2 x=3 or x=3/2`

`:. y=7-x = 7-3/2= 11/2:. D` is `(3/2,11/2)`

Length of altitude `CD` is

`CD = sqrt((x_3-x_4)^2+(y_3-y_4)^2)` or

`CD = sqrt((3-3/2)^2+(7-11/2)^2)= sqrt 4.5 ~~ 2.12` unit

Endpoints of altitude `CD` are `(3,7) and (3/2,11/2)`

Length of altitude `CD` is `2.12` unit [Ans]