A Right Triangle with Side Lengths 5, 12, and 13
A right triangle with side lengths 5, 12, and 13 is a special type of right triangle known as a Pythagorean triple. This is because the sides of this triangle perfectly satisfy the Pythagorean theorem:
a² + b² = c²
Where:
- a and b are the lengths of the two shorter sides (legs) of the right triangle.
- c is the length of the longest side (hypotenuse).
Let's verify this with our triangle:
- a = 5
- b = 12
- c = 13
Substituting these values into the Pythagorean theorem, we get:
- 5² + 12² = 13²
- 25 + 144 = 169
- 169 = 169
The equation holds true, confirming that the triangle with sides 5, 12, and 13 is indeed a right triangle.
Why are Pythagorean Triples Special?
Pythagorean triples are important because they represent whole number solutions to the Pythagorean theorem. This makes them particularly useful in various fields, such as:
- Geometry: For constructing right triangles with specific side lengths.
- Trigonometry: For deriving trigonometric ratios and solving problems related to angles and sides of right triangles.
- Architecture and Engineering: For designing structures and calculating distances and angles.
Other Pythagorean Triples
Besides the well-known 5-12-13 triangle, other common Pythagorean triples include:
- 3-4-5
- 8-15-17
- 7-24-25
These triples can be scaled up by multiplying each side by a common factor. For example, multiplying the 3-4-5 triple by 2 gives us a 6-8-10 Pythagorean triple.
Conclusion
The right triangle with side lengths 5, 12, and 13 is a classic example of a Pythagorean triple. Understanding these triples is crucial for working with right triangles and applying the Pythagorean theorem in various fields.
