Calculating The Hypotenuse Of Triangle DEF A Step-by-Step Guide

In the realm of geometry, understanding the properties of triangles is fundamental. This article delves into the specifics of triangle DEF, where each leg measures 36 units, and aims to determine the length of its hypotenuse. We will explore the underlying mathematical principles, specifically the Pythagorean theorem, to arrive at the correct solution. This exploration is crucial not only for academic purposes but also for practical applications in various fields such as engineering, architecture, and computer graphics. The ability to accurately calculate the dimensions of geometric shapes is a cornerstone of problem-solving in these domains.

Delving into the Properties of Triangle DEF

Triangle DEF presents an interesting scenario in geometry. To accurately determine the length of the hypotenuse, it's important to first understand what type of triangle we are dealing with. The problem states that the triangle has two legs of equal length (36 units each). This key piece of information tells us that triangle DEF is an isosceles triangle, meaning it has at least two sides of equal length. However, to find the hypotenuse, we need to ascertain if it’s a right-angled isosceles triangle. A right-angled triangle is characterized by one angle being exactly 90 degrees. If triangle DEF is indeed a right-angled isosceles triangle, then the two equal legs are the sides adjacent to the right angle, and the hypotenuse is the side opposite the right angle. The hypotenuse is the longest side of a right-angled triangle, and its length can be calculated using the Pythagorean theorem.

The Pythagorean theorem is a fundamental concept in Euclidean geometry that describes the relationship between the sides of a right-angled triangle. The theorem states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). Mathematically, this is expressed as: a² + b² = c², where 'a' and 'b' are the lengths of the legs, and 'c' is the length of the hypotenuse. This theorem provides a powerful tool for solving problems involving right-angled triangles, allowing us to calculate unknown side lengths if we know the lengths of the other two sides. In the context of triangle DEF, if we can confirm it's a right-angled triangle, we can directly apply the Pythagorean theorem to find the length of the hypotenuse.

Before we jump into calculations, it's crucial to visualize the triangle. Imagine a triangle with two sides of equal length meeting at a right angle. The third side, the hypotenuse, stretches across the right angle, forming the longest side of the triangle. Visualizing the triangle helps in understanding the spatial relationships between the sides and angles, making it easier to apply the Pythagorean theorem correctly. Furthermore, understanding the properties of isosceles triangles, such as the fact that the angles opposite the equal sides are also equal, can provide additional insights into the triangle's characteristics. By combining our knowledge of right-angled triangles, isosceles triangles, and the Pythagorean theorem, we can confidently approach the problem of finding the hypotenuse of triangle DEF.

Applying the Pythagorean Theorem to Calculate the Hypotenuse

With a solid understanding of triangle DEF's properties and the Pythagorean theorem, we can now proceed to calculate the length of the hypotenuse. Since we've established that triangle DEF is a right-angled isosceles triangle with legs of 36 units each, we can directly apply the theorem: a² + b² = c². Here, 'a' and 'b' represent the lengths of the legs, which are both 36 units, and 'c' represents the length of the hypotenuse, which is what we want to find. Substituting the values into the equation, we get: 36² + 36² = c². This equation forms the basis for our calculation, and by solving for 'c', we will determine the length of the hypotenuse.

Let's break down the calculation step by step. First, we need to calculate the squares of the leg lengths. 36 squared (36²) is 36 multiplied by 36, which equals 1296. So, the equation becomes: 1296 + 1296 = c². Next, we add the two values on the left side of the equation: 1296 + 1296 = 2592. Now, the equation is simplified to: 2592 = c². To find 'c', the length of the hypotenuse, we need to take the square root of both sides of the equation. The square root of 2592 is approximately 50.91. However, the answer choices are presented in a simplified radical form, so we need to further simplify the square root of 2592. This involves finding the largest perfect square that divides 2592.

To simplify the square root of 2592, we look for perfect square factors. We can start by factoring 2592. We notice that 2592 is divisible by 2, giving us 1296. And 1296 is 36². So we can rewrite 2592 as 2 * 36². Now taking the square root, √2592 = √(2 * 36²) = √2 * √36² = 36√2. This simplification process is crucial for matching our calculated result with the answer choices provided. It demonstrates the importance of not just arriving at a numerical answer, but also expressing it in the correct form. The simplified radical form, 36√2, represents the exact length of the hypotenuse, avoiding any rounding errors that might occur with decimal approximations. This result clearly aligns with one of the given answer choices, solidifying our solution.

Selecting the Correct Answer and Conclusion

Having meticulously calculated the length of the hypotenuse of triangle DEF, we arrive at the solution: 36√2 units. This result directly corresponds to answer choice D. Therefore, the correct answer is D. 36√2 units. This process highlights the importance of understanding the underlying geometric principles, applying the Pythagorean theorem accurately, and simplifying radical expressions to match the expected format of the answer. The journey from understanding the triangle's properties to the final calculation showcases the power of mathematical reasoning and problem-solving skills.

This exploration of triangle DEF and its hypotenuse serves as a valuable exercise in applying geometric principles. By identifying the triangle as a right-angled isosceles triangle, we were able to leverage the Pythagorean theorem to calculate the hypotenuse length. The process of simplifying the square root further demonstrated the importance of algebraic manipulation in arriving at the correct answer in the required format. This example underscores the interconnectedness of different mathematical concepts and their application in solving real-world problems. From architecture to engineering, the ability to accurately determine lengths and dimensions is paramount, making the understanding of these principles essential. In conclusion, the hypotenuse of triangle DEF, with legs of 36 units each, is indeed 36√2 units, reinforcing the significance of geometrical problem-solving in various disciplines.