Introduction to the Pythagorean Theorem

Overview

• Pythagorean Theorem: Applies to right triangles and describes the relationship between the sides of these triangles.
• Named After: Pythagoras, a Greek philosopher and mathematician.

Key Concepts

• Right Triangle: Only applicable to right triangles.
• Sides of a Right Triangle:
  • Hypotenuse: The longest side, opposite the right angle.
  • Legs: The two shorter sides.

Pythagorean Theorem Equation

• Formula: (a^2 + b^2 = c^2)
  • (a) and (b): Represent the lengths of the legs.
  • (c): Always represents the hypotenuse.
• The sum of the squares of the legs equals the square of the hypotenuse.

Example 1

Given

• Right Triangle with:
  • Leg (a = 4) feet
  • Leg (b = 3) feet

Solution

1. Apply Formula: (4^2 + 3^2 = c^2)
2. Calculate:
   • (4^2 = 16)
   • (3^2 = 9)
   • (16 + 9 = 25)
3. Solve for (c):
   • (c^2 = 25)
   • (c = \sqrt{25} = 5)

Visual Representation

• Squaring each side results in squares whose areas fit the Pythagorean theorem.

Example 2

Given

• Right Triangle with:
  • Leg (a = 15) cm
  • Hypotenuse (c = 17) cm

Solution

1. Apply Formula: (15^2 + b^2 = 17^2)
2. Calculate:
   • (15^2 = 225)
   • (17^2 = 289)
   • Subtract (225) from both sides: (b^2 = 64)
3. Solve for (b):
   • (b = \sqrt{64} = 8)

Conclusion

• The Pythagorean Theorem provides a reliable way to calculate side lengths in right triangles.
• Useful for examples and visual representations that reinforce the relationship between the triangle's sides.