Skill Builder: Three Common Right Triangles on the SHSAT/SAT

Right triangle questions show up often on the SHSAT and SAT. Sometimes they’re presented directly, and other times they’re hidden inside questions about rectangles or squares.

For example, you might be given a rectangle that’s 12 feet long and 5 feet wide and asked to find the length of the diagonal. That diagonal is, in fact, the hypotenuse of a right triangle—so this becomes a right triangle problem.

A great way to work faster (and more confidently) is to memorize three common right triangle “triplets" that often appear on the SHSAT and SAT. When you recognize them, you can often avoid doing the full Pythagorean Theorem calculation.

1) The 3–4–5 Triangle

This is one of the most common right triangles you’ll see—either as 3, 4, 5 or as a multiple of it.

Common multiples include:

• 6, 8, 10

• 9, 12, 15

• ¾, 1, 1¼

  
  

Example:If the legs of a right triangle are 0.75 cm and 1 cm, find the perimeter.

Instead of using the Pythagorean Theorem (a² + b² = c²), we recognize that this is, in fact, a 3,4,5 right triangle. The 0.75 and 1 match the ¾ and 1 pattern, so the hypotenuse is 1¼.

Perimeter = ¾ + 1 + 1¼ = 3 cm

2) The 5–12–13 Triangle

Another common one is 5, 12, 13, and it also appears in multiples.

Common multiples include:

• 10, 24, 26

• 15, 36, 39

This triangle is often presented in questions featuring isosceles triangles with a base of 10 inches.

Example: An isosceles triangle with a base of 10 inches and two equal sides that measure 13 inches each. What is the height of this isosceles triangle?

Height from top of triangle to base splits triangle into two right triangles

The height drawn to perpendicularly to the base actually splits the triangle into two right triangles, each with a base of 5 inches. This, in fact, then becomes a 5,12,13 right triangle triplet.

3) The 8–15–17 Triangle

You may also see 8, 15, 17, including larger scaled versions.

Common multiples:

• 400, 750, 850

Final Tip

Memorizing these three right triangle triplets can save you a lot of time on test day. Keep an eye out for them—especially when a problem involves diagonals, rectangles, or squares.