Question 316678: Indicate the equation of the given line in standard form
The line containing the altitude to the hypotenuse of a right triangle whose vertices are P(-1,1), Q(3,5), and R(5,-5)
PLEASE HELP Answer by solver91311(16868) (Show Source):
You really sound desperate. Take a deep breath and relax. This is not as bad as it sounds.
Step 1: Plot the points on a graph.
We are going to use the fact that this triangle is given as a right triangle to simplify this problem considerably. Had it been given as a general triangle we would have to prove the existence of a right angle. Could be done, but it adds to the complexity of the problem.
Clearly, upon inspection, the segment is the hypotenuse of the triangle.
The altitude to any side lies in a line perpendicular to that side. We know that perpendicular lines have slopes that are negative reciprocals, that is to say:
So our next step is to calculate the slope of the line containing the segment . We can do this by applying the slope formula using the coordinates of the given endpoints.
Now we know that the line containing the altitude to the hypotenuse has two features: First, it must have a slope that is the negative reciprocal of the slope of the line containing the hypotenuse, and second, it must pass through point
We have already determined that the slope of the line containing the hypotenuse is . The negative reciprocal is .
Now we have enough information to write the desired equation using the point slope form of the equation of a line:
Just plug in the values for and the coordinates of .
The last step is to put the derived equation into Standard Form. Multiply both sides by 5:
Add to both sides:
This is a perfectly good answer, but I would multiply through by just for the sake of neatness.