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?

Question

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?

James Atkins · Accepted Answer

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

Aurora Ayala · Answer

undefined The altitude going through corner C of the triangle with corners A(2, 5), B(7, 4), and C(2, 3) has endpoints C and the foot of the altitude. The foot of the altitude is the point where the altitude intersects the side opposite to corner C.

To find the foot of the altitude, we need to find the equation of the line passing through A and B, which is the side opposite to corner C. Then we find the perpendicular line passing through C, which will be the altitude. The intersection point of this perpendicular line with the line passing through A and B gives us the foot of the altitude.

First, calculate the slope of the line passing through A and B using the formula:
$$ m = \frac{{y_B - y_A}}{{x_B - x_A}} $$

Then, find the negative reciprocal of this slope to get the slope of the perpendicular line, which is the slope of the altitude.

Next, use the point-slope form of a line to find the equation of the perpendicular line passing through C.

After finding the equation of the perpendicular line, solve it simultaneously with the line passing through A and B to find the intersection point, which is the foot of the altitude.

Finally, you'll have the endpoints of the altitude and can calculate its length using the distance formula.