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

Answer 1

Ella Armstrong

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

#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#

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Answer 2

Maverick Smith

The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To find the orthocenter, you need to find the equations of the altitudes and then solve for their point of intersection. Alternatively, you can use the fact that the orthocenter is the intersection of the perpendicular bisectors of the sides of the triangle. Once you have these equations, you can find the point of intersection, which is the orthocenter.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.