A triangle has corners A, B, and C located at #(4 ,7 )#, #(3 ,5 )#, and #(2 ,1 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2018-01-16T08:02:49

Length of altitude #CF = color(blue)( 0.8944)#

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Slope of AB = #m_(AB) = (y_b - y_a) / (x_b - x_a) = (5-7) / (3-4) = 2#

Slope of AD perpendicular to AB = m_(CF) = - (1/m_(AB) = - (1/2)#

Equation of line altitude AD is

#y - 1 = -(1/2) * (x - 2)#

#2y + x = 4# Eqn (1)

Equation of line segment AB is

#(y - 7) / (5 - 7) = (x - 4) / ( 3 - 4)#

#(y - 7) / -2 = (x - 4) / -1#

#y - 7 = 2x - 8#

#y - 2x = -1# Eqn (2)

Solving Eqns (1) & (2) will give the coordinates of base of altitude F.

#x = 6/5, y = 7/5#

Length of altitude #A CF = sqrt(2-(6/5))^2 + (1-(7/5)^2)#

#AF = color(blue)( 0.8944)#