Question

Determine whether the given lengths are sides of a right triangle. Explain your reasoning. $$5,12,13$$

Short answer

Yes, the lengths 5, 12, and 13 can form a right triangle.

Step-by-step solution

Identifying possible sides of the triangle

Identify the longest length as the potential hypotenuse. In this case, 13 is the possible hypotenuse since it is the longest length.

Apply the Pythagorean theorem

Use the Pythagorean theorem to verify if these lengths can come up with a right triangle. \( a^2 + b^2 = c^2 \) where \( a = 5 \), \( b = 12 \), and \( c = 13 \).

Calculate the sums of the square of the lengths

Calculate \( a^2 + b^2 \) and \( c^2 \) and check if both are equal. \( a^2 + b^2 = 5^2 + 12^2 = 25 + 144 = 169 \) and \( c^2 = 13^2 = 169 \)

Compare the Results

Since 169 = 169, the given lengths indeed form a right triangle.

Key concepts

Understanding the Right Triangle

When it comes to deciphering the properties of geometrical shapes, the right triangle is a fundamental figure to grasp. The defining feature of a right triangle is the presence of a 90-degree angle. In simpler terms, one of the three angles inside the triangle is a perfect square corner.

Considering the sides of a right triangle, the two sides that form the 90-degree angle are referred to as the 'legs' of the triangle while the longest side opposite the right angle is known as the 'hypotenuse'. The significance of the hypotenuse comes into play when we apply formulas such as the Pythagorean theorem.

Let's look at the application in a real-world context. Imagine you're helping a friend build a rectangular frame for a garden bed. To ensure that the corners are right angles, you could measure the sides and use the principles of a right triangle to confirm the accuracy of each corner, facilitating a perfect rectangular shape for the frame.

The Hypotenuse and its Role in the Triangular Harmony

The hypotenuse is more than just the longest side of a right triangle; it is the key to unlocking the mystery of whether a set of three lengths can form a right triangle at all. The placement of the hypotenuse opposite the right angle is strategic. It's akin to being the supporting beam in a bridge, balancing the structure.

In mathematical terms, the hypotenuse intertwines with the Pythagorean theorem to reveal whether three lengths can create a right triangle. It serves as the 'c' in the famous formula: \(a^2 + b^2 = c^2\), representing a verifiable balance between the lengths of the sides when squared.

The crucial aspect here is identifying the hypotenuse correctly as the longest side. Only then can it fulfill its role properly. In practical application, for a construction worker, the hypotenuse might be used to determine if a ladder is placed at the safe and correct angle against a wall.

Squares of Lengths - The Key Calculation

In our journey through geometry, we often encounter the task of squaring numbers, which essentially means multiplying a number by itself. When we talk about the 'squares of lengths' in reference to the Pythagorean theorem, we're highlighting a process central to the theorem's application - calculating the area of squares whose sides are each length of the triangle.

Why square the lengths? It's all about consistent units. When we square each leg and add them up, we should get an area equivalent to the area of the square formed by the hypotenuse. So, in the context of the problem at hand, the lengths 5 and 12 were squared and added to get 169, and the length 13 (the potential hypotenuse) was squared to also get 169. This equality proves, mathematically, that we have a right triangle.

For students or DIY enthusiasts, knowing how to work with squares of lengths can help in tasks such as calculating the area required for materials - be it for a school project or in laying down flooring.