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

1 Answer
sankarankalyanam
Aug 3, 2018

color(maroon)("ortho-centre coordinates " O (73/13, 82/13)ortho-centre coordinates O(7313,8213)

Explanation:

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

![https://www.quora.com/What-is-the-orthocentre-of-a-triangle-when-the-vertices-are-x1-y1-x2-y2-x3-y3](useruploads.socratic.org)

Slope of bar (AB) = m_(AB) = (y_B - y_A) / (x_B - x_A) = (9-3)/ (6-9) = -2¯¯¯¯¯¯AB=mAB=yB−yAxB−xA=9−36−9=−2

Slope of bar(CF) = m_(CF) = - 1/ m(AB) = -1 / -2 = 1/2¯¯¯¯¯¯CF=mCF=−1m(AB)=−1−2=12

Equation of bar(CF) ¯¯¯¯¯¯CF is y - 4 = 1/2 (x - 2)y−4=12(x−2)

2y - x = 72y−x=7 Eqn (1)

Slope of bar (AC) = m_(AC) = (y_C - y_A) / (x_C - x_A) = (4-3)/ (2-9) = -1/7¯¯¯¯¯¯AC=mAC=yC−yAxC−xA=4−32−9=−17

Slope of bar(BE) = m_(BE) = - 1/ m(AC) = -1 / (-1/7) = 7¯¯¯¯¯¯BE=mBE=−1m(AC)=−1−17=7

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

7x - y = 337x−y=33 Eqn (2)

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

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

x = 73/13

y = 164/26 = 82/13

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