A triangle has corners A, B, and C located at #(4 ,7 )#, #(3 ,5 )#, and #(6 ,1 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
bp
Jan 14, 2017

(2,3) ,length of altitude #2sqrt5#

Explanation:

The altitude from the corner C would be perpendicular to line segment AB. Slope of AB would be #(7-5)/(4-3) = 2#. Therefore slope of the perpendicular would be #- 1/2#. Since this perpendicular line passes through corner C (6,1), its equation in point slope form would be
#y-1= -1/2 (x-6)#

Like wise the equation of line AB would be y-7=2(x-4).

The end point of the altitude would be the point of intersection of line AB and the perpendicular line, whose equations have been worked out as above.

From the second equation, y=7+2x-8 = 2x-1. Substitute this for y in the first equation to get #2x-1-1= -1/2 x +3#
Or, #2x +1/2 x= 3+2#
#5x/2 =5 -> x=2# , then y= 2(2)-1=3

Thus the required end point would be (2,3)

Length of the altitude would be #sqrt((6-2)^2 +(1-3)^2)= sqrt20=2sqrt5#

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