A triangle has corners A, B, and C located at #(3 ,1 )#, #(6 ,7 )#, and #(9 ,8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Question

A triangle has corners A, B, and C located at #(3  ,1   )#, #(6  ,7    )#, and #(9 ,8   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Jayden Jackson · Accepted Answer

The end points of altitude are #(9,8) and (7,9) # and
length of altitude is #2.24# unit.

#A(3,1) , B(6,7) , C(9,8)# 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)# #=(7-1)/(6-3) = 6/3=2 :. # Slope of #CD=m_2= -1/m_1= -1/2 # Equation of line #AB# is # y - y_1 = m_1(x-x_1) #or # y- 1 = 2(x-3) or 2x-y = 5 ; (1) # Equation of line #CD# is # y - y_3 = m_2(x-x_3)# or #y- 8 = -1/2(x-9) or 2y-16= -x+9 # or # x+2y=25 ; (2)# Solving equation (1) and (2) we get the co-ordinates of #D(x_4,y_4)#. Mutiplying equation (2) by #2# we get #2x+4y=50 ; (3)# Subtracting equation (1) from equation (3) we get #5y=45 or y=9# # :. x=(5+9)/2=7 :. D# is # (7,9)#. The end points of altitude #CD# are #(9,8) and (7,9)# . Length of altitude #CD# is #CD = sqrt((x_3-x_4)^2+(y_3-y_4)^2) # or #CD = sqrt((9-7)^2+(8-9)^2)= sqrt5 ~~ 2.24# unit [Ans}

Claire Akin · Answer

undefined The endpoint of the altitude going through corner C is (9, 8) because it's perpendicular to the side AB. The length of this altitude can be calculated using the formula for the distance between a point and a line. First, find the equation of line AB using the coordinates of A and B. Then, use the formula for the distance from a point to a line with C as the point and the equation of AB as the line.