A triangle has corners A, B, and C located at #(1 ,3 )#, #(9 ,5 )#, and #(6 ,2 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer

The endpoints are #(93/17, 70/17)#

length#=(9sqrt17)/17=2.18282" " "#units

Explanation:

There is a need to solve for the equation of line from A to B and the line from C perpendicular to line AB. When the two equations are obtained, solve simultaneously for their intersection point . This is the required point D.

Line from A(1, 3) to B(9, 5) using two-point form:

#(y-y_a)=((y_b-y_a)/(x_b-x_a))(x-x_a)#

#(y-3)=((5-3)/(9-1))(x-1)#

#y-3=1/4(x-1)#

#4y-12=x-1#
#x-4y=-11" " "#first equation

To solve for line from C perpendicular to line AB, use slope #=-4#
and point C(6, 2)

Using point-slope form:

#y-y_c=m(x-x_c)#

#y-2=-4(x-6)#

#y-2=-4x+24#

#4x+y=26" " "#second equation

Using first and second equation, solve simultaneously to find the unknow point D.

You should obtain #(x_d, y_d)=(93/17, 70/17)#

length#=sqrt((x_d-x_c)^2+(y_d-y_c)^2)#

length#=sqrt((93/17-6)^2+(70/17-2)^2)#

length#=sqrt((-9/17)^2+(36/17)^2)#

length#=sqrt((9^2/289+(9^2*4^2)/289)#

length#=9*sqrt(17/289)#

length#=9*sqrt(1/17)#

length#=(9sqrt17)/17=2.18282" " "#units

God bless America !