Lecture Notes: Simplifying Radicals and Pythagorean Theorem

Simplifying Radicals

• Simplifying radicals is crucial for understanding right triangles in mathematics.
• Goal: Simplify radicals by finding the square root of perfect squares.
  • Examples of perfect squares:
    • √4 = 2
    • √9 = 3
    • √25 = 5
    • √100 = 10
• Process:
  • Create a factor tree to identify perfect square factors.
  • Example: Simplify √12
    • Factor tree: 12 -> 3 * 4, 4 -> 2 * 2
    • √12 = √(4 * 3) = √4 * √3 = 2√3
  • Example: Simplify √90
    • Factor tree: 90 -> 3 * 30, 30 -> 3 * 10, 10 -> 2 * 5
    • √90 = 3√10
• Technique:
  • Find pairs in the factor tree to simplify to a perfect square.
  • Take one number from the pair outside the square root.*

Pythagorean Theorem

• Statement: For right triangles, the sum of the squares of the legs (a and b) is equal to the square of the hypotenuse (c).
  • Formula: a² + b² = c²
• Pythagorean Triples:
  • Whole number sets satisfying the theorem.
  • Example: (3, 4, 5) and (5, 12, 13).

Examples of Pythagorean Theorem

• Find the Hypotenuse:
  • Given sides a = 20, b = 21
  • a² + b² = c² => 400 + 441 = c² => c² = 841 => c = √841 = 29
• Find a Leg:
  • Given hypotenuse c = 20, one leg b = 8
  • a² + b² = c² => a² + 64 = 400 => a² = 336
  • Simplifying √336 using factor tree:
    • 336 -> 2 * 2 * 2 * 3 * 7
    • √336 = 4√21

Converse of the Pythagorean Theorem

• Statement: If a² + b² = c², then the triangle is a right triangle.
• Example to Test:
  • Triangle with sides 10, 24, and 26.
  • Check: 10² + 24² = 100 + 576 = 676, which equals 26² = 676.
  • Conclusion: It's a right triangle.

Determining Triangle Type

• Right Triangle Test:
  • Example with sides 6, 9, and 12.
  • Check: 6² + 9² = 36 + 81 = 117, which does not equal 12² = 144.
  • Conclusion: Not a right triangle.
• Triangle Types:
  • If c is too big: Obtuse triangle.
  • If c is too small: Acute triangle.