Pythagorean Theorem

TipLearning Objectives
  • Identify right triangles.
  • Apply \(a^2 + b^2 = c^2\).
  • Solve for missing side lengths.

Key Ideas

Right triangle side relationship: \[ a^2 + b^2 = c^2 \] where \(c\) is the hypotenuse.

Right triangle with legs labeled \(a\) and \(b\), and hypotenuse labeled \(c\).

Common Problem Types

Solving for Hypotenuse

Example: legs = 6 and 8 → \(c = 10\).

Solving for a Leg

Example: \(c = 13\), \(a = 5\).

Identifying a Right Triangle

Check if sides satisfy the theorem.

Coordinates

Distance formula: \[ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]

Strategies

  • Always confirm which side is hypotenuse.
  • Use perfect squares when possible.
  • For coordinate problems, set up the distance formula carefully.

Worked Examples

Example 1

Legs: 7 and 24 →
\[ c=\sqrt{49+576}=25. \]

Example 2

Sides: 10, 24, 26 → check right triangle:
\[10^2 + 24^2 = 26^2\]
\[100 + 576 = 676\]

WarningCommon Mistakes
  • Using Pythagorean Theorem on non-right triangles.
  • Treating the longest side as a leg.
  • Forgetting to square both sides.

Practice Problems

  1. Legs: 9, 12 → find hypotenuse.
  2. Hypotenuse = 17, leg = 15 → find other leg.
  3. Check if 8, 15, 17 is a right triangle.
  1. \(c = 15\)
  2. \(b = 8\)
  3. \(8^2 + 15^2 = 17^2\): 64+225=289 → yes

Summary

  • Pythagorean Theorem applies only to right triangles.
  • Hypotenuse is always opposite the right angle.
  • Check for perfect-square triples.
  • Always identify the right angle first.