A triangle has corners A, B, and C located at #(5 ,6 )#, #(3 ,9 )#, and #(4 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2017-12-02T11:36:44

Length of altitude passing through point C = 3.0309#

Equation of side AB
#(y-6)/(9-6) = (x-5)/(3-5)#
#-2(y-6) = 3(x-5)#
#3x + 2y = 27#. Eqn (1)

Let Slope of side AB be ‘m’
#m = (9-6) / (3-5) = -(3/2)#
Slope of perpendicular line to AB is #= -(1/m) = -(1/(-(3/2))) = 2/3#

Eqn of Altitude to AB passing through point C is
#(y-2) =( 2/3)(x-4)#
#3y - 6 = 2x - 8#

#2x - 3y = 2#. Eqn (2)

Solving Eqns (1) & (2) we get the base of the altitude passing through point C.

Solving the two equations, we get
#13y = 48, y = 3(9/13)#

#13x = 85, x = 6(7/13)#

Length of the altitude passing through point C
#= sqrt((4-6(7/13))^2 + (2-(3(9/13))^2)#
#= sqrt(2(7/13)^2 + 1(9/13)^2) = sqrt(6.4438 + 2.8639)#

Length of Altitude passing through point C = 3.0509#

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