A triangle has corners A, B, and C located at #(1 ,3 )#, #(3 ,5 )#, and #(4 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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Answer 1 · 2018-03-01T12:14:09

The end points of altitude are #(4,2) and (2,4)# and length of altitude is #2.83# unit.

#A(1,3) , B(3,5) , C(4,2)#

Let #CD# be the altitude going through #C# touches #D# on line

#AB#. #C# and #D# are the endpoints of altitude #CD; CD# is

perpendicular on #AB#. Slope of #AB= m_1= (y_2-y_1)/(x_2-x_1)#

#=(5-3)/(3-1) =1 :. # Slope of #CD=m_2= -1/m_1= -1/1=-1 #

Equation of line #AB# is # y - y_1 = m_1(x-x_1) #or

# y- 3 =1(x-1) or x-y = -2 ; (1) #

Equation of line #CD# is # y - y_3 = m_2(x-x_3)# or

#y- 2 = -1(x-4) or x+y=6 ; (2) # Solving equation

(1) and (2) we get the co-ordinates of #D(x_4,y_4)#. Adding

equation (1) and equation(2) we get #2x=4 or x=2#

#y=6-x or y=4 :. D# is # (2,4)#. The end points of altitude are

#CD# is #(4,2) and (2,4)# . Length of altitude #CD# is

#CD = sqrt((x_3-x_4)^2+(y_3-y_4)^2) # or

#CD = sqrt((4-2)^2+(2-4)^2)= sqrt8 ~~ 2.83# unit [Ans]