Introduction to the Pythagorean Theorem

Overview

• The Pythagorean Theorem is fundamental in mathematics, particularly in geometry and trigonometry.
• It is used to calculate distances between points and is essential for solving right triangle problems.

Right Triangle Basics

• A right triangle has one angle measuring 90 degrees, known as the right angle.
• The side opposite the right angle is the longest side, called the hypotenuse.
• Right triangles have two other sides, referred to as the legs of the triangle.

The Pythagorean Theorem Formula

• The formula is expressed as: ( A^2 + B^2 = C^2 )
  • ( A ) and ( B ) are the legs of the triangle.
  • ( C ) is the hypotenuse.
• It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Example Problems

Example 1

1. Given a right triangle with sides 3 and 4, find the hypotenuse.
2. Identify the hypotenuse as the side opposite the right angle.
3. Apply the theorem:
   • ( 3^2 + 4^2 = C^2 )
   • ( 9 + 16 = 25 )
   • ( C^2 = 25 )
   • ( C = 5 ) (Taking the positive square root because distances are positive)

Example 2

1. Given a right triangle with a hypotenuse of 12 and one leg of 6, find the other leg.
2. Identify the hypotenuse and apply the theorem:
   • ( A^2 + 6^2 = 12^2 )
   • ( A^2 + 36 = 144 )
   • ( A^2 = 108 )
   • Solve for ( A ) by taking the square root:
     • ( A = \sqrt{108} )
     • Simplify ( \sqrt{108} = 6\sqrt{3} )

Simplifying Radicals

• Factoring 108: ( 108 = 2 \times 2 \times 3 \times 3 \times 3 )
• Simplified radical form: ( \sqrt{108} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3} )

Important Notes

• Always correctly identify the hypotenuse before applying the theorem.
• Positive square roots are used in the context of distance calculations.
• Simplifying radicals involves breaking down numbers into their prime factors and grouping into perfect squares.

Applications

• The Pythagorean Theorem is not only crucial in mathematics but also applicable in various fields such as physics and engineering for distance and measurement calculations.