Pythagorean Triples & Special Right Triangles

TipLearning Objectives
  • Recognize common Pythagorean triples.
  • Use 45-45-90 and 30-60-90 triangle rules.
  • Scale special triangles for quick side-length computation.

Key Ideas

Common Pythagorean Triples

  • 3-4-5
  • 5-12-13
  • 7-24-25
    Any multiple (e.g., 6-8-10) is also a triple.

Special Right Triangles

45-45-90

Sides follow: \[ x,\; x,\; x\sqrt{2} \]

30-60-90

Sides follow: \[ x,\; x\sqrt{3},\; 2x \] (short, long, hypotenuse)

Diagrams of the 45°–45°–90° triangle and the 30°–60°–90° triangle with standard side ratios labeled.

Common Problem Types

Using Triples

Identify when 3-4-5 structure appears.

Scaling Triples

Multiply all sides equally.

Using 45-45-90

Leg → multiply by \(\sqrt{2}\) to get hypotenuse.

Using 30-60-90

Short side is key.

Choosing Fast Method

Use triple or special triangle before Pythagorean Theorem.

Strategies

  • Look for multiples of known triples.
  • Identify the shortest side in 30-60-90 triangles → sets everything.
  • Memorize side ratios.

Worked Examples

Example 1 — Triple

Sides: 9, 12 → identify as 3×3-4×3 → hypotenuse = 15.

Example 2 — 45-45-90

Leg = 10 → hypotenuse = \(10\sqrt2\).

Example 3 — 30-60-90

Short side = 5 → long side = \(5\sqrt3\), hypotenuse = 10.

WarningCommon Mistakes
  • Mixing up which side is the “short side” in 30-60-90.
  • Forgetting to scale all sides equally.
  • Using Pythagorean Theorem unnecessarily.

Practice Problems

  1. 3-4-5 scaled by 7 → find sides.
  2. 45-45-90 with hypotenuse 8 → find legs.
  3. 30-60-90 with long leg 9√3 → find short leg.
  1. 21, 28, 35
  2. Legs = \(4\sqrt2\)
  3. Long leg = \(x\sqrt3 = 9\sqrt3 → x = 9\)

Summary

  • Triples save time.
  • 45-45-90 → \(x, x, x\sqrt2\)
  • 30-60-90 → \(x, x\sqrt3, 2x\)
  • Short side of 30-60-90 is opposite 30°.
  • Look for hidden triples!