The Verification of a Right Triangle with Side Lengths 3 and 4

Let's explore the fascinating world of right triangles and how to verify if a triangle with side lengths of 3 and 4 (and an implied hypotenuse of 5) is indeed a right triangle. We will utilize multiple methods to ensure a thorough understanding of the properties of this triangle.

Method 1: Pythagorean Theorem

The Pythagorean theorem is one of the most well-known theorems in geometry and is specifically designed to verify the presence of a right triangle. According to the theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Let's apply this to our triangle with side lengths of 3, 4, and 5.

Calculation: 32 42 9 16 25

And, 52 25

Since both sides are equal, the triangle with side lengths 3, 4, and 5 is indeed a right triangle.

Method 2: Exploring Pythagorean Triples

It's noteworthy that (3, 4, 5) is a classic example of a Pythagorean triple, where the squares of the two smaller sides sum up to the square of the largest side.

Calculation: 32 42 9 16 25 52

This method not only confirms the right triangle but also introduces the concept of Pythagorean triples.

Method 3: Using the Cosine Rule

The cosine rule, also known as the Law of Cosines, can be used to confirm if a triangle is a right triangle. This rule states that c2 a2 b2 - 2ab cos(C), where C is the angle opposite to side c. For a right triangle, cos(90°) 0. Applying this to our triangle:

Calculation:

c2 a2 b2 - 2ab cos(C)

52 32 42 - 2 × 3 × 4 × cos(90°)

25 9 16 - 0

25 25

Since the equation holds true, the angle opposite to the hypotenuse (5) is 90°, confirming that the triangle is indeed a right triangle.

Additional Insights

In a right triangle, the longest side is called the hypotenuse, which is always opposite the right angle. The other two sides are known as the legs, with the one adjacent to the angle being called the adjacent side or base, and the one opposite to the angle being called the opposite or perpendicular height. Understanding these terms can be useful in solving various problems related to right triangles.

The verification of a right triangle with side lengths 3 and 4 using the Pythagorean theorem, Pythagorean triples, and the cosine rule showcases the versatility and interconnectivity of geometric principles. These methods not only help in understanding the properties of triangles but also provide a solid foundation for more complex geometric and trigonometric problems.