Pythagorean Theorem: Applications and Proofs

• The Pythagorean Theorem is a fundamental concept in geometry that relates the sides of a right triangle
• It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs)
• Algebra is used to solve it

Pythagorean Theorem

• The theorem is expressed by the equation: a² + b² = c², where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the legs
• The theorem applies only to right triangles, which are triangles containing one angle of 90 degrees
• The hypotenuse is always the longest side of a right triangle and is opposite the right angle
• The legs are the two sides that form the right angle

Applications of the Pythagorean Theorem

• Finding the Length of a Side: If the lengths of two sides of a right triangle are known, the Pythagorean Theorem can be used to find the length of the third side
• Determining if a Triangle is a Right Triangle: If the lengths of all three sides of a triangle are known, the Pythagorean Theorem can be used to determine whether the triangle is a right triangle. If a² + b² = c², then the triangle is a right triangle
• Real-World Applications: The Pythagorean Theorem has numerous practical applications in fields such as construction, navigation, and surveying. For example, it can be used to calculate the length of a diagonal beam in a building or to determine the distance between two points on a map

Algebraic Manipulation of the Pythagorean Theorem

• Solving for a Leg: The Pythagorean Theorem can be rearranged to solve for either of the legs (a or b) if the lengths of the hypotenuse and the other leg are known. The formulas are: a = √(c² - b²) and b = √(c² - a²)
• Solving for the Hypotenuse: To solve for the hypotenuse (c), take the square root of both sides of the original equation: c = √(a² + b²)
• Simplifying Radicals: In many cases, the lengths of the sides of a right triangle will involve square roots (radicals). Algebraic techniques can be used to simplify these radicals to obtain the most accurate and simplified answer

Example Problems

• Finding the Hypotenuse: Given a right triangle with legs of lengths 3 and 4, find the length of the hypotenuse
  • Using the Pythagorean Theorem: c² = 3² + 4² = 9 + 16 = 25
  • Taking the square root of both sides: c = √25 = 5. The length of the hypotenuse is 5
• Finding a Leg: Given a right triangle with a hypotenuse of length 13 and one leg of length 5, find the length of the other leg
  • Using the Pythagorean Theorem: a² + 5² = 13²
  • Rearranging to solve for a: a² = 13² - 5² = 169 - 25 = 144
  • Taking the square root of both sides: a = √144 = 12. The length of the other leg is 12
• Determining if a Triangle is a Right Triangle: A triangle has sides of lengths 7, 24, and 25. Determine if it is a right triangle
  • Using the Pythagorean Theorem: 7² + 24² = 49 + 576 = 625
  • Comparing to the square of the longest side: 25² = 625
  • Since 7² + 24² = 25², the triangle is a right triangle

Common Pythagorean Triples

• A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy the Pythagorean Theorem, a² + b² = c²
• Knowing common Pythagorean triples can save time when solving problems
• Examples of common Pythagorean triples
  • (3, 4, 5): This is the most well-known Pythagorean triple. Multiples of this triple (e.g., (6, 8, 10), (9, 12, 15)) are also Pythagorean triples
  • (5, 12, 13)
  • (8, 15, 17)
  • (7, 24, 25)

Advanced Applications and Extensions

• Distance Formula: The distance formula in coordinate geometry is derived from the Pythagorean Theorem. It is used to find the distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane: d = √((x₂ - x₁)² + (y₂ - y₁)² )
• 3D Pythagorean Theorem: In three-dimensional space, the Pythagorean Theorem can be extended to find the diagonal of a rectangular prism. If the dimensions of the prism are a, b, and c, then the length of the diagonal d is given by: d = √(a² + b² + c²)
• Trigonometry: The Pythagorean Theorem is foundational to trigonometry. Trigonometric functions such as sine, cosine, and tangent are defined in terms of the ratios of the sides of a right triangle