The Pythagorean Theorem: Complete Guide
Theorem Statement
In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the other two sides.
Where:
- c is the length of the hypotenuse (the side opposite the right angle)
- a and b are the lengths of the other two sides (the legs)
History and Background
Named after the Greek mathematician Pythagoras (570-495 BCE), this theorem has been known for over 2,500 years. Evidence suggests that ancient civilizations like the Babylonians understood the relationship even earlier.
Visual Representation
The theorem can be visually understood by drawing squares on each side of a right triangle:
Key Insight: The area of the square on the hypotenuse (c²) equals the sum of the areas of the squares on the other two sides (a² + b²).
Important Pythagorean Triples
Pythagorean triples are sets of three positive integers a, b, and c that satisfy the equation a² + b² = c².
| Triple (a, b, c) | Verification |
|---|---|
| (3, 4, 5) | 3² + 4² = 9 + 16 = 25 = 5² |
| (5, 12, 13) | 5² + 12² = 25 + 144 = 169 = 13² |
| (8, 15, 17) | 8² + 15² = 64 + 225 = 289 = 17² |
| (7, 24, 25) | 7² + 24² = 49 + 576 = 625 = 25² |
| (9, 40, 41) | 9² + 40² = 81 + 1600 = 1681 = 41² |
Tip: Any multiple of a Pythagorean triple is also a Pythagorean triple. For example, (6, 8, 10) is a multiple of (3, 4, 5).
Examples and Applications
Example 1: Finding the Hypotenuse
A ladder is placed against a wall. The base of the ladder is 6 meters from the wall, and the ladder reaches 8 meters up the wall. How long is the ladder?
Solution:
Let's denote the length of the ladder as c, the distance from the wall as a = 6 m, and the height up the wall as b = 8 m.
Using the Pythagorean theorem: c² = a² + b²
c² = 6² + 8²
c² = 36 + 64
c² = 100
c = 10
Therefore, the ladder is 10 meters long.
Example 2: Finding a Leg
A rectangular television screen has a diagonal of 50 inches and a width of 40 inches. What is the height of the screen?
Solution:
Let's denote the height as a, the width as b = 40 inches, and the diagonal as c = 50 inches.
Using the Pythagorean theorem: c² = a² + b²
50² = a² + 40²
2500 = a² + 1600
a² = 2500 - 1600
a² = 900
a = 30
Therefore, the height of the screen is 30 inches.
Example 3: Checking if a Triangle is Right-Angled
Determine whether a triangle with sides of lengths 9, 40, and 41 units is right-angled.
Solution:
For a triangle to be right-angled, the square of the length of the longest side should equal the sum of the squares of the other two sides.
The longest side is 41, so we check if 41² = 9² + 40²
41² = 1681
9² + 40² = 81 + 1600 = 1681
Since 1681 = 1681, the triangle is right-angled.
Example 4: Distance Between Two Points
Find the distance between the points (1, 2) and (4, 6) in a coordinate plane.
Solution:
The distance formula is derived from the Pythagorean theorem:
distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
distance = √[(4 - 1)² + (6 - 2)²]
distance = √[3² + 4²]
distance = √[9 + 16]
distance = √25
distance = 5
Therefore, the distance between the two points is 5 units.
Example 5: Navigation and Bearings
A ship sails 24 miles east and then 10 miles north. How far is the ship from its starting point?
Solution:
The east distance is 24 miles (a), and the north distance is 10 miles (b).
Using the Pythagorean theorem to find the direct distance c:
c² = a² + b²
c² = 24² + 10²
c² = 576 + 100
c² = 676
c = 26
Therefore, the ship is 26 miles from its starting point.
Different Ways to Solve Pythagorean Theorem Problems
1. Direct Application
This is the most common method, where we directly substitute values into the formula c² = a² + b².
Example: Find the hypotenuse of a right triangle with legs of lengths 5 and 12.
c² = 5² + 12²
c² = 25 + 144
c² = 169
c = 13
2. Rearranging to Find a Leg
When we know the hypotenuse and one leg, we can rearrange the formula to find the other leg:
a² = c² - b²
Example: Find the length of one leg of a right triangle if the hypotenuse is 17 and the other leg is 8.
a² = c² - b²
a² = 17² - 8²
a² = 289 - 64
a² = 225
a = 15
3. Using the Converse
The converse of the Pythagorean theorem states that if the squares of two sides of a triangle equal the square of the third side, then the triangle is a right triangle.
Example: Determine if a triangle with sides 20, 21, and 29 is a right triangle.
Check if 29² = 20² + 21²
841 = 400 + 441
841 = 841
Since the equation holds, the triangle is a right triangle.
4. Using Special Right Triangles
Certain right triangles have specific angle-side relationships:
30°-60°-90° Triangle
If the shortest leg is x, then:
- The hypotenuse is 2x
- The longer leg is x√3
45°-45°-90° Triangle
If both legs are x, then:
- The hypotenuse is x√2
Example: In a 45°-45°-90° triangle with legs of length 10, find the hypotenuse.
hypotenuse = 10√2 ≈ 14.14
5. Algebraic Approach with Pythagorean Triples
All primitive Pythagorean triples can be generated with the formulas:
- a = m² - n²
- b = 2mn
- c = m² + n²
Where m and n are positive integers with m > n, and m and n are coprime (no common factors) and not both odd.
Example: Generate a Pythagorean triple using m = 4 and n = 1.
a = 4² - 1² = 16 - 1 = 15
b = 2(4)(1) = 8
c = 4² + 1² = 16 + 1 = 17
So (15, 8, 17) is a Pythagorean triple.
Check: 15² + 8² = 225 + 64 = 289 = 17²
6. Geometric Proof Method
This involves visualizing the relationship using area calculations:
- Draw a right triangle with sides a, b, and hypotenuse c
- Draw squares on each side
- Demonstrate that the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides