# How to Find the Right Angle to Two Points

If you’re given two points on the coordinate plane*,* you can find the locus of points that creates a right angle to those two points. By using the following steps, you’ll soon see an interesting pattern that may, in fact, surprise you.

Here’s a problem to solve: Given points *P* and *R,* what’s the locus of points *Q* such that angle *PQR* is a right angle?

**Identify a pattern.**

This pattern may be a bit tricky to find, but if you start with points*P*and*R*and try to find a few points*Q*that make a right angle with*P*and*R,*you’ll probably begin to see a pattern emerging, as shown here.See the pattern? The

*Q*points are beginning to form a circle with diameter*PR*. This makes sense if you think about the inscribed-angle theorem: In a circle with line*PR*as its diameter, semicircular arc*PR*would be 180 degrees, so all inscribed angles*PQR*would be one-half of that, or 90 degrees.**Look outside the pattern.**

Nope, nothing to add here. Any point*Q*inside the circle you identified in Step 1 creates an*obtuse*angle with*P*and*R*(or a straight angle), and any point*Q*outside the circle creates an*acute*angle with*P*and*R*(or a zero degree angle). All the right angles are on the circle.**Look inside the pattern.**

Bingo. See what points have to be excluded? It’s the given points*P*and*R.*If*Q*is at the location of either given point, all you have left is a segment (*QR*or*PQ*), so you no longer have the three distinct points you need to make an angle.**Draw the locus and describe it in words.**The final figure shows the locus, and the caption gives its description. Note the hollow dots at

*P*and*R,*which indicate that those points aren’t part of the solution.