Introduction to the Pythagorean Theorem

Overview

• The Pythagorean theorem is used to describe the relationship between the sides of right triangles.
• Named after Pythagoras, a Greek philosopher and mathematician.
• Applicable only to right triangles.

Triangle Components

• Right Triangle: A triangle with one angle measuring 90 degrees.
• Hypotenuse: The side opposite the right angle; the longest side of the triangle.
• Legs: The two shorter sides that form the right angle.

Pythagorean Theorem Equation

• Expressed as ( a^2 + b^2 = c^2 ).
  • ( a ) and ( b ) represent the lengths of the legs.
  • ( c ) represents the length of the hypotenuse.
• This equation allows us to calculate the missing side length if two sides are known.

Example 1

• Given: Two legs of a triangle measuring 4 feet and 3 feet.
• Find: The hypotenuse.
• Calculation:
  • ( 4^2 + 3^2 = c^2 )
  • ( 16 + 9 = c^2 )
  • ( 25 = c^2 )
  • ( c = \sqrt{25} = 5 )
• Result: The hypotenuse is 5 feet.
• Visual Representation: Squaring each side creates squares whose areas demonstrate the theorem (16 + 9 = 25).

Example 2

• Given: One leg measuring 15 centimeters and hypotenuse measuring 17 centimeters.
• Find: The missing leg.
• Calculation:
  • ( 15^2 + b^2 = 17^2 )
  • ( 225 + b^2 = 289 )
  • ( b^2 = 289 - 225 )
  • ( b^2 = 64 )
  • ( b = \sqrt{64} = 8 )
• Result: The missing leg is 8 centimeters.

Key Points

• The theorem holds true for all right triangles.
• The visual representation of squaring the sides illustrates the theorem’s principle.
• The theorem can be used to find a missing side given the other two sides are known.

This introduction to the Pythagorean theorem provides foundational understanding and examples applicable to right triangles. Further practice with various triangle measurements can strengthen comprehension and application.