A triangle has corners A, B, and C located at #(3 ,8 )#, #(7 ,5 )#, and #(2 ,9 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2018-01-03T07:25:02

Coordinates of CF (2,9) , (47/25, 221/25))

Length of Altitude CF = #color(purple)(0.2)#

Equation of side AB is
#(y-y1)/(y2-y1)=(x-x1)/(x2-x1)#
#(y-8)/(5-8)=(x-3)/(7-3) #
#4y - 32 = -3x + 9#
#color(brown)(3x + 4y = 41, (Equation 1))#

Slope of AB m1#=(y2-y1)/(x2-x1)=(5-8)/(7-3)=-(3/4)#
Slope of altitude CF
#m2=-1/(m1) = -1/(-3/4)= 4/3#

Equation of altitude CF
#(y-y3)=m2(x-x3)#
#(y-9) = (4/3)(x-2)#
#(3y - 27 = 4x - 8#
#color(brown)(3y - 4x = 19, (Equation 2))#

By solving #color(brown)(Equations 1 & 2)#, we get point F
#x = 47/25, y = 221/25#
Coordinates of #F (47/25, 221/25)#

Length of altitude CF

#CF=sqrt(((47/25)-2)^2+((221/25)-9)^2)#

#CF=sqrt((-3/25)^2+(-4/25)^2)= color(brown)(1/5 = 0,2)#