What is the orthocenter of a triangle with corners at $(3 ,1 )$ , $(4 ,5 )$ , and (2 ,7 )#?

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Answer 1 · 2018-01-05T12:02:39

Orthocenter coordinates $color(red)((-1/3), (-7/3)$

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Slope of $BC = m_(bc) = (y_b - y_c) / (x_b - x_c) = (5-7)/(4-2) = -1$

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

Equation of AD is
$y - 1 = (1) * (x - 3)$

$color(red)(-x + y = -2)$ Eqn (1)

Slope of $AB = m_(AB) = (y_a - y_b) / (x_a - x_b) = (1-5)/(3-4) = 4$

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

Equation of CF is
$y - 7 = (4) * (x - 2)$

$color(red)(-4x + y = -1)$ Eqn (2)

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

Solving the two equations,
$x = -(1/3), y = -(7/3)$

Coordinates of orthocenter $color(purple)(O (-1/3, -7/3))$

What is the orthocenter of a triangle with corners at (3 ,1 )(3 ,1 ), (4 ,5 )(4 ,5 ), and (2 ,7 )#?