The Right Angle Rule: Pythagorean Theorem
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². This single formula unlocks distance calculations, structural measurements, and navigation problems that appear in construction, engineering, and everyday geometry.
Components of the Pythagorean Theorem
This section covers the key applications of the theorem:
- Finding the Hypotenuse: Given the two legs a and b, calculate c = √(a² + b²).
- Finding a Leg: Given the hypotenuse c and one leg, calculate the missing leg: a = √(c² - b²).
- Pythagorean Triples: Sets of three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13, and 8-15-17.
- Real-World Applications: Using the theorem to find distances, check right angles, and solve practical measurement problems.
Examples of the Pythagorean Theorem
Finding the Hypotenuse Examples
- A right triangle has legs of 3 and 4. Using a² + b² = c²: 9 + 16 = 25, so c = √25 = 5.
- A right triangle has legs of 6 and 8. Calculate 36 + 64 = 100, so c = √100 = 10.
- A right triangle has legs of 5 and 12. Calculate 25 + 144 = 169, so c = √169 = 13.
Finding a Leg Examples
- A right triangle has hypotenuse 10 and one leg 6. Calculate a = √(100 - 36) = √64 = 8.
- A right triangle has hypotenuse 13 and one leg 5. Calculate a = √(169 - 25) = √144 = 12.
- A ladder 15 feet long leans against a wall with its base 9 feet from the wall. The height it reaches is √(225 - 81) = √144 = 12 feet.
Pythagorean Triples Examples
- The triple 3-4-5 can be scaled: multiply by 2 for 6-8-10, by 3 for 9-12-15, and so on.
- Check if 7-24-25 is a triple: 49 + 576 = 625 and 25² = 625 ✓.
- Check if 5-11-12 is a triple: 25 + 121 = 146 but 12² = 144, so no — not a right triangle.
Applications Examples
- A rectangular field is 40 meters by 30 meters. The diagonal distance across it is √(1,600 + 900) = √2,500 = 50 meters.
- A TV screen measures 36 inches wide and 27 inches tall. The diagonal is √(1,296 + 729) = √2,025 = 45 inches.
- Two hikers start at the same point. One walks 8 km north and the other walks 6 km east. The straight-line distance between them is √(64 + 36) = √100 = 10 km.