Finding The Hypotenuse Of A 45-45-90 Triangle With 14 Cm Legs
When delving into the fascinating world of geometry, the 45-45-90 triangle stands out as a special right triangle with unique properties. Understanding these properties allows us to quickly solve problems involving these triangles, bypassing complex calculations. This article will explore how to determine the hypotenuse length when each leg of a 45-45-90 triangle measures 14 cm. We will dissect the characteristics of this special triangle, apply the relevant theorem, and arrive at the solution, ensuring a comprehensive understanding of the underlying principles.
Understanding 45-45-90 Triangles
To effectively tackle this problem, a solid grasp of the 45-45-90 triangle is crucial. This triangle is a right-angled triangle, meaning it has one angle measuring 90 degrees. What sets it apart is that the other two angles are both 45 degrees. This unique angle configuration results in specific relationships between the sides, making it an isosceles right triangle. Isosceles triangles, by definition, have two sides of equal length. In a 45-45-90 triangle, these equal sides are the legs, which are the sides adjacent to the right angle. The side opposite the right angle is the hypotenuse, which is always the longest side of a right triangle. The 45-45-90 triangle theorem is the cornerstone for solving problems involving these triangles. This theorem states that the sides are in a specific ratio: 1 : 1 : √2. This ratio means that if the legs have a length of 'x', then the hypotenuse has a length of 'x√2'. This proportional relationship allows us to easily calculate the hypotenuse if we know the length of a leg, and vice versa. Recognizing this ratio is the key to efficiently solving problems involving 45-45-90 triangles. By internalizing this relationship, you can avoid using the Pythagorean theorem in many cases, saving time and effort. The theorem simplifies calculations and provides a direct path to the solution. In essence, the 45-45-90 triangle theorem provides a shortcut for finding side lengths, making it an invaluable tool in geometry and trigonometry. Understanding this theorem is not just about memorizing a ratio; it's about comprehending the underlying geometric principles that govern these special triangles. This knowledge empowers you to approach problems with confidence and accuracy, fostering a deeper appreciation for the elegance and interconnectedness of mathematical concepts. The 45-45-90 triangle is a fundamental building block in geometry, and mastering its properties is essential for further exploration in mathematics.
Applying the 45-45-90 Triangle Theorem
Now, let's apply the 45-45-90 triangle theorem to our specific problem. We are given that each leg of the triangle measures 14 cm. The theorem states that the sides of a 45-45-90 triangle are in the ratio 1 : 1 : √2. This implies that if the length of each leg is 'x', the hypotenuse will be 'x√2'. In our case, 'x' is 14 cm. To find the length of the hypotenuse, we simply substitute 14 cm for 'x' in the expression 'x√2'. This gives us 14√2 cm. Therefore, the length of the hypotenuse is 14√2 cm. This calculation demonstrates the power and simplicity of the 45-45-90 triangle theorem. By recognizing the triangle as a special right triangle, we could directly apply the ratio and avoid more complex methods like the Pythagorean theorem, which would involve squaring the leg lengths, adding them, and then taking the square root. This direct application of the theorem not only saves time but also reduces the chance of errors. The 45-45-90 triangle theorem provides a clear and concise path to the solution. Understanding and memorizing this theorem is essential for anyone studying geometry and trigonometry. It is a fundamental tool that unlocks a range of problems involving these special triangles. The ability to quickly apply the theorem showcases a strong understanding of geometric principles and problem-solving skills. This example highlights the importance of recognizing patterns and relationships in mathematics. By identifying the 45-45-90 triangle, we could leverage its unique properties to efficiently determine the hypotenuse length. This approach exemplifies the beauty of mathematical reasoning and its ability to simplify complex situations. Ultimately, the 45-45-90 triangle theorem is a valuable asset in any mathematical toolkit, providing a straightforward method for solving problems involving these special right triangles.
Solution and Answer
Based on our application of the 45-45-90 triangle theorem, we've determined that the length of the hypotenuse is 14√2 cm. Looking at the options provided:
A. 7 cm B. 7√2 cm C. 14 cm D. 14√2 cm
We can clearly see that option D (14√2 cm) is the correct answer. This confirms our calculation and understanding of the theorem. The other options are incorrect because they do not reflect the proportional relationship between the legs and the hypotenuse in a 45-45-90 triangle. Option A (7 cm) is significantly smaller than the leg length, which is impossible for a hypotenuse. Option B (7√2 cm) is half the correct answer, suggesting a misunderstanding of the ratio. Option C (14 cm) is equal to the leg length, which is also incorrect as the hypotenuse must be longer than either leg in a right triangle. The correct answer, 14√2 cm, accurately reflects the ratio of sides in a 45-45-90 triangle. It's important to note that √2 is an irrational number, approximately equal to 1.414. Therefore, 14√2 cm is approximately 14 * 1.414 cm, which is roughly 19.8 cm. This value is greater than the leg length (14 cm), as expected for the hypotenuse. This exercise reinforces the importance of understanding the properties of special right triangles and applying the relevant theorems correctly. The ability to identify the correct answer and justify it based on the theorem demonstrates a solid grasp of geometric concepts. This problem-solving process not only provides the solution but also strengthens the understanding of the underlying mathematical principles. In conclusion, the correct answer is D, 14√2 cm, which is derived directly from the 45-45-90 triangle theorem and confirms the proportional relationship between the sides.
Conclusion
In conclusion, determining the hypotenuse of a 45-45-90 triangle when the legs measure 14 cm is a straightforward application of the 45-45-90 triangle theorem. This theorem provides a direct relationship between the leg lengths and the hypotenuse, allowing us to quickly calculate the hypotenuse as 14√2 cm. Understanding and applying this theorem is crucial for efficiently solving problems involving these special right triangles. The 45-45-90 triangle serves as a fundamental concept in geometry and trigonometry. Its unique properties simplify calculations and provide a clear understanding of side relationships. Mastering this theorem not only helps in solving specific problems but also builds a strong foundation for more advanced mathematical concepts. The problem-solving process, as demonstrated in this article, highlights the importance of recognizing patterns and applying the appropriate theorems. By identifying the triangle as a 45-45-90 triangle, we could leverage its specific properties to arrive at the solution with ease. This approach exemplifies the power of mathematical reasoning and its ability to simplify complex situations. Furthermore, this exercise reinforces the significance of accuracy and attention to detail in mathematical calculations. The correct application of the theorem and the correct identification of the answer are essential for achieving the desired result. In summary, the 45-45-90 triangle theorem is a valuable tool in geometry, providing a direct and efficient method for finding side lengths in these special triangles. The solution to this problem, 14√2 cm, underscores the importance of understanding and applying mathematical principles correctly. This knowledge empowers us to confidently tackle similar problems and deepen our appreciation for the beauty and elegance of mathematics. The ability to solve this type of problem is a testament to a solid understanding of geometric concepts and problem-solving skills, paving the way for further exploration in the world of mathematics.
