Question

What is the orthocenter of a triangle with corners at `(3 ,6 )`, `(4 ,2 )`, and (5 ,7 )#?

Answer 1

Orthocenter of the triangle `color(purple)(O (17/9, 56/9))`

Explanation:

Slope of `BC = m_(bc) = (y_b - y_c) / (x_b - x_c) = (2-7)/4-5) = 5`

Slope of `AD = m_(ad) = - (1/m_(bc) = - (1/5)`

Equation of AD is
`y - 6 = -(1/5) * (x - 3)`

`color(red)(x + 5y = 33)` Eqn (1)

Slope of `AB = m_(AB) = (y_a - y_b) / (x_a - x_b) = (6-2)/(3-4) = -4`

Slope of `CF = m_(CF) = - (1/m_(AB) = - (1/-4) = 4`

Equation of CF is
`y - 7 = (1/4) * (x - 5)`

`color(red)(-x + 4y = 23)` Eqn (2)

Solving Eqns (1) & (2), we get the orthocenter `color(purple)(O)` of the triangle

Solving the two equations,
`x = 17/9, y = 56/9`

Coordinates of orthocenter `color(purple)(O (17/9, 56/9))`