Geometry and Special Right Triangles

Special Right Triangles

45-45-90 Triangle

• Angles: One 90° angle, two 45° angles
• Side Lengths:
  • Opposite each 45° angle: length a
  • Hypotenuse: a√2
• Example Calculations:
  • If one side is 3, hypotenuse = 3√2
  • If hypotenuse is 8, side length = 8/√2 (can be rationalized to 4√2)

30-60-90 Triangle

• Angles: One 90° angle, one 30° angle, one 60° angle
• Side Lengths:
  • Opposite 30° angle: length a
  • Opposite 60° angle: length a√3
  • Hypotenuse: 2a
• Example Calculations:
  • If side opposite 30° is 5, hypotenuse = 10 and opposite 60° = 5√3
  • If hypotenuse is 30, side opposite 30° = 15 and opposite 60° = 15√3
  • If side opposite 60° is 8, side opposite 30° = 8/√3 (rationalized to 8√3/3) and hypotenuse = 16√3/3

Key Concepts

• Use the Pythagorean Theorem to derive side lengths: a² + b² = c².
• Rationalizing the denominator: multiply numerator and denominator by the same radical to avoid radicals in the denominator.
• Understanding derivations helps in remembering and reproducing the formulas during tests.

Tips

• Always verify calculations with the Pythagorean Theorem.
• Practice with different given values to become comfortable with derivations.
• Remember that not all results will be nice whole numbers.

Final Notes

• These triangles often appear in geometry problems, making these results very useful.
• If formulas are forgotten, understanding the derivation process helps in quickly reproducing them.

Feel free to ask questions or provide comments for further clarification.