Question

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

Answer 1

`color(maroon)("ortho-centre coordinates " O (73/13, 82/13)`

Explanation:

`A (9,3), B(6,9), C(2,4)`

Slope of `bar (AB) = m_(AB) = (y_B - y_A) / (x_B - x_A) = (9-3)/ (6-9) = -2`

Slope of `bar(CF) = m_(CF) = - 1/ m(AB) = -1 / -2 = 1/2`

Equation of `bar(CF)` is `y - 4 = 1/2 (x - 2)`

`2y - x = 7` Eqn (1)

Slope of `bar (AC) = m_(AC) = (y_C - y_A) / (x_C - x_A) = (4-3)/ (2-9) = -1/7`

Slope of `bar(BE) = m_(BE) = - 1/ m(AC) = -1 / (-1/7) = 7`

Equation of `bar(BE)` is `y - 9 = 7 (x - 6)`

`7x - y = 33` Eqn (2)

Solving Eqns (1) and (2), we get the ortho-centre coordinates `O(x,y)`

`cancel(2y) - x + 14x - cancel(2y) = 7 + 66`

`x = 73/13`

`y = 164/26 = 82/13`