A triangle has corners A, B, and C located at #(2 ,3 )#, #(5 ,8 )#, and #(4 , 2 )#, 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 #(2   ,3   )#, #(5  ,8    )#, and #(4 ,  2   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Maverick Smith · Accepted Answer

The end points are #=(71/34,107/34)# and the length of the altitude is #=2.23#

The triangle's corners are

#A=(2,3)#

#B=(5,8)#

#C=(4,2)#

The slope of the line #AB# is #m=(8-3)/(5-2)=5/3#

The equation of line #AB# is

#y-3=5/3(x-2)#

#y-3=5/3x-10/3#

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

#mm'=-1#

The slope of the line perpendicular to #AB# is #m'=-3/5#

The equation of the altitude through #C# is

#y-2=-3/5(x-4)#

#y-2=-3/5x+12/5#

#y=-3/5x+22/5#................................#(2)#

Solving for #x# and #y# in equations #(1)# and #(2)#, we get

#5/3x-1/3=-3/5x+22/5#

#3/5x+5/3x=1/3+22/5#

#34/15x=71/15#

#x=71/34#

#y=-3/5*71/34+22/5=535/170=107/34#

The end points of the altitude is #=(71/34,107/34)#

The altitude's duration is

#=sqrt((4-71/34)^2+(2-107/34)^2)#

#=sqrt((65/34)^2+(-39/34)^2)#

#=sqrt(5746)/34#

#=2.23#

Claire Akin · Answer

undefined To find the endpoints and length of the altitude going through corner C of the triangle with vertices A(2, 3), B(5, 8), and C(4, 2), you can follow these steps:

1. Determine the slope of the line segment AB using the coordinates of points A and B.
2. Find the equation of the line passing through A and B using the point-slope form.
3. Determine the perpendicular line to AB passing through point C, as it represents the altitude.
4. Find the intersection point of the altitude line and the line segment AB to get the endpoints of the altitude.
5. Calculate the distance between these endpoints to find the length of the altitude.

Let's proceed with the calculations.