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

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Answer 1 · 2016-10-23T21:06:50

The endpoints are #(9,8) and (365/53, 3192/371)#

The distance is #~~ 2.2#

Use the point-slope form of the equation of a line to write the equation of the line through points A and B:

#y - 2 = (2 - 9)/(5 - 7)(x - 5)#

#y - 2 = (-7)/(-2)(x - 5)#

#y - 2 = (7)/(2)x - 35/2#

#y = (7)/(2)x - 31/2# [1]

We need the above form and the standard form:

#2y - 7x + 31 = 0# [2]

The slope of the altitude through point is the negative reciprocal of the slope in equation [1], #-2/7#

Use the point-slope form of the equation of a line to find the equation of the altitude through point C:

#y - 8 = -2/7(x - 9)#

#y - 8 = -2/7x + 18/7#

#y = -2/7x + 74/7# [3]

To find the x coordinate of the other endpoint, subtract equation 3 from equation [1]

#y - y = (7)/(2)x + 2/7x - 31/2 - 74/7#

#0 = (53)/(14)x - 365/14#

#(53)/(14)x = 365/14#

#x = 365/53#

To find the y coordinate of the other endpoint, substitute the above into equation [3]:

#y = -2/7(365/53) + 74/7#

#y = 3192/371#

Use equation [2] to find the length of the altitude:

#d = |2(8) - 7(9) + 31|/sqrt(2^2 + (-7)^2)#

#d ~~ 2.2#