How to Find the Right Angle to Two Points - dummies

How to Find the Right Angle to Two Points

By Mark Ryan

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?

  1. 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.

    Identifying “Q” points that form right angles with P and R.

    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.

  2. 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.
  3. 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.
  4. Draw the locus and describe it in words.
    Given points P and R, the locus of points Q such that angle PQR is a right angle is a circle with diameter PR, minus points P and R.

    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.