A triangle has corners A, B, and C located at #(1 ,1 )#, #(6 ,8 )#, and #(7 ,4 )#, 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 #(1  ,1   )#, #(6  ,8    )#, and #(7 ,4   )#, respectively. What are the endpoints and length of the altitude going through corner C?

Sadie Adams · Accepted Answer

Endpoint: #(329/74, 431/74)#
Height: #27/sqrt(74)#

Here is a graph.

What we want to know is the endpoint(#H#) and length(#CH#) of the altitude.

[Step1: determine the line #AB#]
The slope of line #AB# is #(8-1)/(6-1)=7/5# and the equation is
#y-1=7/5(x-1)#
#y=7/5x-2/5#.

[Step2: determinte the line #CH#]
If the two lines with slope #m_1# and #m_2# cross at the right angle, #color(red)(m_1m_2=-1)# is needed.
Thus, the slope of line CH is #-1/(7/5)= -5/7#.
Its equation is #y-4=-5/7(x-7)#. i.e. #y=-5/7x+9#.

[Step3: calculate the endpoint and height]
Solving the system equations
#y=7/5x-2/5#
#y=-5/7x+9#
and the endpoint #H# is #H(329/74, 431/74)# or, approximately #(4.4459,5.8243)#.

The length of CH is calculated below:
#CH=sqrt((7-329/74)^2+(4-431/74)^2)#
#=27/sqrt(74)#
#=3.1387…#

Theodore Abad · Answer

undefined The endpoints of the altitude going through corner C are (7, 4) and the foot of the altitude, which is the point where the altitude intersects the side opposite corner C. To find the foot of the altitude, first, determine the slope of the line passing through points A and B. Then, find the perpendicular slope to this line, which will be the slope of the altitude. Finally, using the slope of the altitude and the coordinates of point C, find the equation of the line passing through C with this slope. Solve this equation along with the equation of the line passing through points A and B to find the foot of the altitude. Once you have the foot of the altitude, you can calculate the length of the altitude using the distance formula between point C and the foot of the altitude.