How to Determine if Three Lengths Form a Right Triangle

Mathematics, particularly geometry, frequently deals with the properties of triangles. A right triangle is a special type of triangle that has one angle measuring 90 degrees. One of the most well-known theorems related to right triangles is the Pythagorean theorem.

The Basics of Right Triangles

A right triangle consists of three sides. The longest side, which is opposite the right angle, is called the hypotenuse. The other two sides are called the legs. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, this is expressed as:

a2 b2 c2

Determining Right Triangles with the Pythagorean Theorem

To check if three given lengths form a right triangle, you can apply the Pythagorean theorem. Here's a step-by-step method:

Identify the longest side, which is the hypotenuse (c). Square the hypotenuse. Add the squares of the other two sides. Check if the sum from step 2 is equal to the square from step 3.

Example 1: 3, 4, 5

Let's take the triangle with sides 3, 4, and 5.

The longest side is 5 (hypotenuse). 52 25. 32 42 9 16 25. 25 25, so this is a right triangle.

Alternative Method: Product of Alternate Sides

There is another, less conventional method to determine if a triangle is a right triangle by examining the relationship between the sides. If the product of two alternate sides plus 1 is equal to the square of the middle side, then the triangle is a right triangle.

Example 2: 3, 4, 5

In the 3, 4, 5 triangle:

The alternate sides are 3 and 5. The product of the alternate sides is 3 * 5 15. The square of the middle side is 42 16. 16 - 1 15. 15 15, so this is a right triangle.

Additional Examples

Example 3: 5, 12, 13 52 25 122 52 144 25 169 169 132, so this is a right triangle. Example 4: 7, 24, 25 72 49 242 72 576 49 625 625 252, so this is a right triangle.

Conclusion

The Pythagorean theorem is a powerful tool for determining whether a triangle is a right triangle. By applying the theorem and using the methods discussed, you can quickly and easily check if the given lengths form a right triangle. These methods are not only useful in academic settings but also have practical applications in various fields, such as architecture, engineering, and physics.

Remember that the relationship between the sides can be simply checked using the Pythagorean theorem, but understanding the concept and its applications provides a deeper appreciation of geometry and its relevance in everyday life.