A triangle has corners A, B, and C located at #(7 ,1 )#, #(4 ,3 )#, and #(5 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2018-02-07T04:20:44

End points of the altitude C (5,8), D (31/13, 53/13)#

Length of altitude #CD = **4.715** #

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Slope AB #m_(AB) = (3-1) / (4-7) = -2/3

Slope of CD perpendicular to AB #m_(CD) = -1 / (-2/3) = 3/2#

Equation of line segment AB

#(y-1) / (3-1) = (x - 7) / (4-7)#

#-3 * (y-1) = 2 * (x - 7)#

#3y + 2x = 17# Eqn (1)

Equation of altitude CD through point C is

#(y - y_C = m_(CD) (x - x_C)#

#y - 8 = (3/2) (x - 5)#

#2y - 16 = 3x - 15#

#2y - 3x = 1# Eqn (2)

Solving Eqns (1), (2) we get the coordinates of D, base of altitude CD.

#D (31/13, 53/13)#

Length of altitude #CD = sqrt((5-(31/13))^2 + (8-(53/13))^2 = 4.715#