A triangle has corners A, B, and C located at #(2 ,5 )#, #(7 ,4 )#, and #(6 ,3 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2016-10-14T18:04:18

The end point is #(81/13, 54/13)#

The length of the altitude is: #a = sqrt(234)/13#

The slope, m, of the line that opposite C is:

#m = (5 - 4)/(2 - 7) = 1/-5 = -1/5#

Use the point-slope form of the equation of a line to write the equation of line c (opposite angle C):

#y - 5 = (-1/5)(x - 2)#

#y - 5 = (-1/5)x + 2/5#

#y = (-1/5)x + 27/5#

The slope, n, of the altitude perpendicular to line c is:

#n = -1/m = -1/(-1/5) = 5#

Use the point-slope form of the equation of a line to write the equation for the altitude line:

#y - 3 = 5(x - 6)

#y = 5x - 27#

Because #y = y#, we can set the right sides equal, to find the x coordinate of their intersection:

#5x - 27 = (-1/5)x + 27/5#

#25x - 135 = -x + 27#

#26x = 162#

#x = 81/13#

y = 54/13

The end point is #(81/13, 54/13)#

The length of the altitude is:

#a = sqrt((6 - 81/13)^2 + (3 - 54/13)^2)#

#a = sqrt((-3/13)^2 + (-15/13)^2)#

#a = sqrt(234)/13#