Enter Two Known Sides
Triangle Diagram
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the three sides of a right triangle. In any right triangle (where one angle is exactly 90 degrees), the theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This relationship is expressed mathematically as:
Where:
• c represents the length of the hypotenuse
• a and b represent the lengths of the other two sides
Practical Applications
The Pythagorean Theorem has numerous real-world applications:
- Construction: Ensuring corners are perfectly square (90 degrees)
- Navigation: Calculating shortest distances between points
- Architecture: Designing structures with right angles
- Physics: Resolving force vectors and calculating distances
- Computer Graphics: 3D rendering and spatial calculations
Example Calculations
Finding the Hypotenuse (c)
If you know the lengths of legs a and b, you can find the hypotenuse c:
Example: If a = 3 and b = 4
Finding a Leg (a or b)
If you know the hypotenuse and one leg, you can find the other leg:
Example: If c = 10 and a = 6
Algebraic Proofs
The Pythagorean Theorem has been proven in many ways throughout history. Here are two algebraic proofs that demonstrate the validity of the theorem:
Proof 1: Using Area Comparison
In this arrangement:
- Four identical right triangles (with legs a and b) are arranged around a central square (with side c)
- This forms a larger square with side length (a + b)
- The area of the large square = (a + b)²
- The area of the central square = c²
- The combined area of the four triangles = 4 × (½ab) = 2ab
Therefore:
Proof 2: Alternative Area Comparison
In this arrangement:
- Four identical right triangles are arranged to form a larger square with side c
- The enclosed space forms a smaller square with side length (b - a)
- The area of the large square = c²
- The area of the small square = (b - a)²
- The combined area of the four triangles = 4 × (½ab) = 2ab
Therefore:
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