Find a Right Triangle’s Missing Side Using Triangle Families

By Mark Ryan

If you can’t immediately see which Pythagorean triple family a triangle belongs to, don’t worry: you can always use the following step-by-step method to pick the family and find the missing side.

For example, say you’re given the following tricky triangle:

You can use a ratio to figure out what family this triangle belongs to.
  1. Take the two known sides and make a ratio (either in fraction form or colon form) of the smaller to the larger side.


  2. Reduce this ratio to whole numbers in lowest terms.


    (With a calculator, this step is a snap because many calculators have a function that reduces fractions to lowest terms.)

  3. Look at the fraction from Step 2 to spot the particular triangle family.

    The numbers 4 and 5 are part of the 3-4-5 triangle, so you’re dealing with the 3 : 4 : 5 family.

  4. Divide the length of a side from the given triangle by the corresponding number from the family ratio to get your multiplier (which tells you how much the basic triangle has been blown-up or shrunk).

    Use the length of the hypotenuse from the given triangle (because working with a whole number is easier) and divide it by the 5 from the 3 : 4 : 5 ratio.


  5. Multiply the third family number (the number you don’t see in the reduced fraction in Step 2) by the result from Step 4 to find the missing side of your triangle.


    That’s the length of side p; and that’s a wrap.

You may be wondering why you should go through all this trouble when you could just use the Pythagorean Theorem. Good point. The Pythagorean Theorem is easier for some triangles (especially if you’re allowed to use your calculator). However, this triple triangle technique can come in handy. Take your pick.