# Finding possible coordinates of a right triangle given one side

You are given the coordinates of two vertices a right triangle (not specified whether or not they describe the hypotenuse), and asked which of a set of possible coordinates could describe the third vertex: Possible answers: A (0,0), B (9,6), C (2,4), D (10,7), E (4,9).

What's the easiest way to solve this? I can think of two approaches:

• For each of the answer options, find the lengths of the sides of the resulting triangle and see if they're in a Pythagorean relationship.
• For each of the answer options, find the slopes of all three sides and see if any two of them are negative inverse to each other.

These are both rather time-consuming. Is there a simpler way to solve this?

• Plotting the points and the "side" $\ AB \$ may suggest which points are candidates and which can be eliminated as possibilities; that will save some calculation. Working with slopes will probably be faster.
– user882145
Jun 22, 2021 at 1:01

If you notice that the slope of the segment between your two points is $$-\frac{1}{2}$$ then any point on the line through $$(3,7)$$ with slope $$2$$ will work. That line has equation $$\frac{y-7}{x-3}=2\implies y=2x+1$$. We can immediately see that $$(4,9)$$ is on that line so that will work.