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

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Answer 1 · 2018-04-15T11:39:23

Endpoints of altitude #CD# are #(3,7) and (3/2,11/2)#
Length of altitude #CD# is # 2.12# unit

#A(5,2) , B(2,5) , C(3,7)#

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)#

#=(5-2)/(2-5) = 3/(-3)= -1 :. # Slope of #CD=m_2= -1/m_1= 1 #

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

# y-2 = -1(x-5) or x +y = 7 ; (1) #

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

#y- 7 = 1(x-3) or x -y = -4 (2) # Solving equation (1) and

(2) we get the co-ordinates of #D#

Adding equation (1) and (2) we get #2 x=3 or x=3/2#

#:. y=7-x = 7-3/2= 11/2:. D # is #(3/2,11/2)#

Length of altitude #CD# is

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

#CD = sqrt((3-3/2)^2+(7-11/2)^2)= sqrt 4.5 ~~ 2.12# unit

Endpoints of altitude #CD# are #(3,7) and (3/2,11/2)#

Length of altitude #CD# is # 2.12# unit [Ans]