A triangle has corners A, B, and C located at #(1 ,7 )#, #(7 ,4 )#, and #(2 , 8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2016-10-02T19:58:19

The end points are (2, 8) and (4.6, 5.2), The length is #a ~~ 3.82#

Going from point A to B we go right 6 and down 3, therefore, the slope is #-1/2#

The equation of the line through points A and B is:

#y - 7 = -1/2(x -1)#

#y - 14/2 = -1/2x + 1/2#

#y = 15/2 - 1/2x#

The altitude through point C will perpendicular to the above line, thereby, making its slope the negative reciprocal of #-1/2#. The slope is 2.

The equation of the line through point C is:

#y - 8 = 2(x - 2)#

#y - 8 = 2x - 4#

#y = 2x + 4#

Set the right sides of the two equations equal and then solve for the x coordinate of their intersection:

#15/2 - 1/2x = 2x + 4#

#15/2 - 1/2x = 4/2x + 8/2#

#23/2 = 5/2x#

#x = 23/5#

#x = 4.6#

#y = 15/2 - 1/2(4.6)#

#y = 5.2#

The length, a, of the altitude is:

#a = sqrt((4.6 - 2)² + (5.2 - 8)²)#

#a ~~ 3.82#