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

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Answer 1 · 2017-03-27T20:04:43

The end points of the altitude are #(2,4) and (86/25,127/25)#
The length is #9/5#

Find the slope and the equation of the line segment AB

#m_(AB) = (3 - 7)/(5-2)#

#m_(AB) = -4/3#

Use the slope and one of the points to find the equation of the line segment AB:

#y = -4/3(x - 2)+7" [1]"#

Use the slope of AB to compute the slope of the Altitude.

#m_("altitude") = -1/m_(AB)#

#m_("altitude") = -1/(-4/3)#

#m_("altitude") = 3/4#

Use the slope and point C to find the equation of the altitude:

#y = 3/4(x - 2)+4" [2]"#

Use equations [1] and [2] to find the x coordinate of the point of intersection:

#-4/3(x - 2)+7 = 3/4(x - 2)+4#

#-4/3(x - 2)- 3/4(x - 2) = -3#

#16(x - 2)+ 9(x - 2) = 36#

#25(x - 2) = 36#

#x - 2 = 36/25#

#x = 50/25+36/25#

#x = 86/25#

Use either equation to find the y coordinate; I will use [2]:

#y = 3/4(36/25)+4#

#y = 27/25+100/25#

#y = 127/25#

The end points of the altitude are #(2,4) and (86/25,127/25)#

The length of the altitude is the distance between the two points:

#d = sqrt((86/25-2)^2+(127/25-4)^2#

#d = sqrt(36^2+27^2)/25#

#d = sqrt(2025)/25#

#d = 9/5#