Question

What is the orthocenter of a triangle with corners at `(6 ,2 )`, `(3 ,7 )`, and (4 ,9 )#?

Answer 1

Coordinates of orthocenter `color(blue)(O (16/11, 63/11))`

Explanation:

Slope of BC `= m_a = (9-7)/(4-3) = 2`

Slope of AD `= -1/m_a = -1/2`

Equation of AD is

`y - 2 = -(1/2)(x - 6)`

`2y - 4 = -x + 6`

`2y + x = 10` Eqn (1)

Slope of CA `= m_b = (9-2) / (4-6) = -(7/2)`

Slope of BE `= - (1/m_b) = 2/7`

Equation of BE is

`y - 7 = (2/7) (x - 3)`

`7y - 49 = 2x - 6`

`7y - 2x = 43` Eqn (2)

Solving Eqns (1), (2) we get the coordinates of ‘O’ the orthocenter

`color(blue)(O (16/11, 63/11))`

Confirmation:
`Slope of AB = m_c = (7-2)/(3-6) = -(5/3)`
`Slope of AD = -1/m_c = 3/5`
Equation of CF is
`y - 9 = (3/5)(x - 4)`
`5y - 3x = 33` Eqn (3)
Solving Eqns (1), (3) we get
`color(blue)(O (16/11, 63/11))`