Understanding the Geometry of Right Triangles: Pythagorean Theorem and Special Ratios
Right triangles are a fundamental type of triangle with widespread applications across various fields.
The unique properties of right triangles play a significant role in trigonometry, a branch of mathematics dedicated to the study of these triangular configurations. This article will delve into one of the most renowned theorems associated with right triangles—The Pythagorean Theorem.
Right Triangle
A right triangle is a triangle with one of its interior angles being a right angle. Below are some of the features common to all right triangles.
- The side opposite to the right angle is always the longest side and is known as the hypotenuse.
- The sum of the two non-right angle interior angles is 90 degrees.
- A right triangle with the two non-right angles being congruent (each 45 degrees) is an isosceles right triangle. Here the two perpendicular sides are of identical length.
The Pythagorean theorem describes the relationship between the three sides of a right triangle.
The Pythagorean Theorem
The Pythagorean theorem, describing the relationship between the three sides of a right triangle, is attributed to the ancient Greek mathematician and philosopher Pythagoras. While Pythagoras is credited with its discovery in the 6th century BCE, evidence suggests its use in Egypt, Babylon, and China dating back to the 20th century BCE.
There are several proofs for the Pythagoras theorem. Below is a visual proof of the theorem:
When placing three squares with side lengths corresponding to the three sides (a, b, and c) of a right triangle, Pythagoras proved that the area of the largest square was equivalent to the sum of the other two squares. The Pythagoras theorem therefore derives:
c² = a² + b²
Here, c is the hypotenuse (longest side of a right triangle), and a, b are the other two sides, and their labels can be interchanged.
Applications of the Pythagorean Theorem
The Pythagorean theorem finds practical applications, with its most straightforward use being the determination of the missing side of a right triangle. It allows for the calculation of the third side when two sides of a right triangle are known. The example below shows a right triangle with a height of 8 inches and a base of 15 inches. The hypotenuse can be determined as shown:
Some real-life applications that extend from this use case are the following:
- Finding the distance between two points on the Cartesian plane.
- Determining the length of a staircase required to reach a specific height.
Special Ratios
For practical purposes and to minimize calculations people have identified certain ratios that fit the Pythagorean theorem. The ratio is built from Pythagorean triples. Pythagorean triples are three numbers that fit into the Pythagorean theorem. For example 3,4, and 5.
52 = 32 + 42
A triangle drawn with its sides being numbers in a Pythagorean triple would always be a right triangle.
Construction workers leverage Pythagorean triples to achieve right-angle corners without sophisticated tools. By using ratios like 3:4:5, they can create a right triangle with sides measuring 3ft, 4ft, and 5ft respectively, enabling the creation of square corners on a large scale, like constructing foundations or creating perfectly rectangular flower beds.
You incorporate a string and create a triangle with the three sides being 3ft, 4ft, and 5ft respectively, the angle that is opposite of the hypotenuse will always be a right angle. Interestingly, this mechanism was first used in ancient Egypt, when they wanted to separate rectangular plots from large lands.
Any 3 sides that are in the ratio 3:4:5 will yield the same result. For example, a triangle with sides 600 m, 800 m, and 1000 m respectively will be a right triangle, as 600:800:1000 simplifies to the ratio 3:4:5. This is why 3:4:5 is a special ratio. There are infinitely many Pythagorean triples, and thus many special ratios such as:
- 5: 12: 13
- 7: 24: 25
- 8: 15: 17
- 5: 12: 13
- 20: 99: 101