Calculate Hypotenuse Length Of A 45-45-90 Triangle With 14 Cm Legs

This article delves into the fascinating world of 45-45-90 triangles, a special type of right triangle that exhibits unique properties. Specifically, we will explore how to determine the length of the hypotenuse when the length of each leg is known. In this case, we are given a 45-45-90 triangle where each leg measures 14 cm, and our objective is to calculate the length of the hypotenuse. Understanding the relationships within these triangles is crucial in various fields, including geometry, trigonometry, and even practical applications like construction and engineering.

Understanding 45-45-90 Triangles

To effectively calculate the hypotenuse, it's essential to first grasp the fundamental characteristics of 45-45-90 triangles. These triangles are characterized by two equal angles of 45 degrees and one right angle of 90 degrees. This unique angle configuration leads to a specific relationship between the sides: the two legs are congruent (equal in length), and the hypotenuse is always ${\sqrt{2}}$ times the length of a leg. This relationship is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). In a 45-45-90 triangle, if we denote the length of each leg as 'a', then the hypotenuse 'c' can be calculated as follows:

${c^2 = a^2 + a^2}$

${c^2 = 2a^2}$

${c = a\sqrt{2}}$

This formula is a cornerstone in solving problems related to 45-45-90 triangles, and it allows us to quickly determine the hypotenuse if we know the length of a leg, or vice versa. The consistent ratio between the sides makes these triangles particularly useful in various geometric and trigonometric calculations. For instance, when dealing with squares, diagonals naturally form 45-45-90 triangles, making this knowledge invaluable in architectural design and spatial reasoning. Furthermore, understanding this relationship simplifies many complex problems, allowing for efficient and accurate solutions. Recognizing the inherent properties of 45-45-90 triangles not only enhances problem-solving skills but also provides a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

Applying the 45-45-90 Triangle Theorem

In this specific problem, we are given that each leg of the 45-45-90 triangle measures 14 cm. To find the length of the hypotenuse, we can directly apply the formula derived from the 45-45-90 triangle theorem, which states that the hypotenuse is equal to the length of a leg multiplied by ${\sqrt{2}}$. Mathematically, this can be represented as:

Hypotenuse = Leg Length * ${\sqrt{2}}$

Substituting the given leg length of 14 cm into the formula, we get:

Hypotenuse = 14 cm * ${\sqrt{2}}$

Therefore, the length of the hypotenuse is ${14\sqrt{2}}$ cm. This straightforward application of the theorem highlights the efficiency and practicality of understanding the specific relationships within special right triangles. The 45-45-90 triangle theorem is not merely a formula to memorize; it's a powerful tool that simplifies geometric calculations. By recognizing the inherent ratios within these triangles, we can bypass more complex methods and arrive at solutions quickly and accurately. This theorem is particularly valuable in situations where time is a constraint, such as standardized tests or real-world applications where quick calculations are necessary. Moreover, the ability to apply this theorem demonstrates a deeper understanding of geometric principles, moving beyond rote memorization to a conceptual grasp of mathematical relationships. The clarity and precision offered by this theorem underscore its importance in both theoretical and practical contexts.

Step-by-Step Solution

To ensure clarity and understanding, let's break down the solution into a step-by-step process. This approach is beneficial for those who prefer a structured method and helps to avoid potential errors. The step-by-step solution not only reinforces the application of the formula but also provides a clear pathway for solving similar problems in the future. Each step is designed to build upon the previous one, creating a logical progression that leads to the final answer. This methodical approach is particularly useful for students learning geometry, as it encourages a systematic way of thinking and problem-solving.

1. Identify the Triangle Type: The problem explicitly states that we are dealing with a 45-45-90 triangle. This is the crucial first step, as it allows us to apply the specific properties associated with this type of triangle. Recognizing the triangle type is the key to unlocking the correct solution method. Without this initial identification, we might be tempted to use more complex methods that are unnecessary in this case.
2. Recall the 45-45-90 Triangle Theorem: The theorem states that in a 45-45-90 triangle, the hypotenuse is ${\sqrt{2}}$ times the length of a leg. This is the fundamental principle that guides our calculation. Recalling this theorem is essential for applying the correct formula and achieving an accurate result. The theorem provides a direct relationship between the sides, making the calculation straightforward and efficient.
3. Identify the Given Information: We are given that each leg of the triangle measures 14 cm. This is the information we will use to calculate the length of the hypotenuse. Clearly identifying the given information is a critical step in any problem-solving process. It ensures that we are using the correct values in our calculations and avoids confusion.
4. Apply the Formula: Using the 45-45-90 triangle theorem, we multiply the leg length (14 cm) by ${\sqrt{2}}$. This step directly applies the principle we recalled in Step 2. The application of the formula is a mechanical step, but it is crucial for arriving at the correct answer. Accurate substitution of values into the formula is essential for success.
5. Calculate the Hypotenuse Length: 14 cm * ${\sqrt{2}}$ = ${14\sqrt{2}}$ cm. This calculation gives us the length of the hypotenuse. The final calculation is the culmination of all the previous steps. It provides the numerical answer to the problem and completes the solution process. The result, ${14\sqrt{2}}$ cm, is the exact length of the hypotenuse.

By following these steps, we can confidently determine the length of the hypotenuse in a 45-45-90 triangle when the leg length is known. This systematic approach not only aids in solving this specific problem but also equips us with a valuable problem-solving strategy for future challenges in geometry and beyond.

Choosing the Correct Answer

Based on our calculations, the length of the hypotenuse is ${14\sqrt{2}}$ cm. Now, let's examine the given options to identify the correct answer. This is a critical step in the problem-solving process, as it ensures that we select the answer that accurately reflects our calculations. Choosing the correct answer involves comparing our calculated result with the provided options and identifying the match. This step is often overlooked but is crucial for avoiding careless errors and ensuring that our hard work culminates in the right selection.

A. 7 cm - This option is incorrect. It does not account for the ${\sqrt{2}}$ factor required by the 45-45-90 triangle theorem. Selecting this option would indicate a misunderstanding of the theorem or a calculation error.

B. ${7\sqrt{2}}$ cm - This option is also incorrect. While it includes the ${\sqrt{2}}$ factor, it uses half the leg length in the calculation. This suggests a possible error in applying the formula or a confusion about which side lengths to use.

C. 14 cm - This option is incorrect. It represents the length of the leg, not the hypotenuse. Choosing this option would demonstrate a failure to correctly apply the 45-45-90 triangle theorem.

D. ${14\sqrt{2}}$ cm - This option is the correct answer. It matches our calculated hypotenuse length and correctly applies the 45-45-90 triangle theorem. Selecting this option confirms that we have accurately understood and applied the relevant geometric principles.

Therefore, the correct answer is D. ${14\sqrt{2}}$ cm. This process of elimination and comparison ensures that we not only arrive at the correct answer but also understand why the other options are incorrect. This deeper understanding strengthens our grasp of the concepts and enhances our problem-solving skills.

Conclusion

In conclusion, by understanding the properties of 45-45-90 triangles and applying the relevant theorem, we can efficiently determine the length of the hypotenuse when the leg length is known. In this specific case, with each leg measuring 14 cm, the hypotenuse was calculated to be ${14\sqrt{2}}$ cm. This exercise not only reinforces the importance of recognizing special right triangles but also highlights the power of applying geometric principles to solve practical problems. The 45-45-90 triangle theorem is a valuable tool in geometry, offering a direct and efficient way to calculate side lengths in these specific triangles. Mastering this theorem and its applications is essential for students and professionals alike, providing a solid foundation for more advanced geometric concepts. Furthermore, the step-by-step approach outlined in this article serves as a model for tackling other mathematical challenges, emphasizing the importance of a systematic and logical problem-solving strategy. By understanding the underlying principles and applying them methodically, we can confidently navigate a wide range of geometric problems and develop a deeper appreciation for the elegance and precision of mathematics.