A triangle has corners A, B, and C located at #(2 ,5 )#, #(7 ,4 )#, and #(2 ,3 )#, respectively. What are the endpoints and length of the altitude going through corner C?

1 Answer
May 14, 2016

The altitude basis coordinates over the side #AB# are #(2.38462, 4.92308)# and the altitude length is #1.96116#

Explanation:

Given #A, B, C# the altitude goes perpendicularly from #C# to the side #AB#. The altitude direction is obtained choosing a direction perpendicular to the segment #s=lambda A + (1-lambda)B# for #0 le lambda le 1#. The segment direction is given by the vector #v=B-A#.

A direction perpendicular to #v# is easily obtained, changing the #v# components and changing a sign for one of them.

Example:
The vector #v^T# perpendicular to #v = (v_x,v_y)# has components
#v^T = (v_y,-v_x)#
So the altitude straight is given by
#h = C + v^T mu#
The intersection point is obtained solving for #(lambda,mu)# the system
#lambda A + (1-lambda)B= C + v^T mu#
The altitude basis in side #AB# is #(2.38462, 4.92308)# and the altitude length is #1.96116#