Finding The Leg Length Of A 45-45-90 Triangle With Hypotenuse 4 Cm

The hypotenuse of a 45-45-90 triangle is 4 cm. What is the length of one leg?

The $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, also known as an isosceles right triangle, is a fundamental geometric shape that frequently appears in mathematics and various real-world applications. Understanding the relationships between its sides is crucial for solving problems related to trigonometry, geometry, and engineering. In this article, we will explore the properties of this special triangle and demonstrate how to calculate the length of its legs when the hypotenuse is known. Specifically, we will address the problem: "The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 4 cm. What is the length of one leg of the triangle?"

A $45^{\circ}-45^{\circ}-90^{\circ}$ triangle is a right triangle with two acute angles each measuring 45 degrees. This unique configuration results in several special properties that simplify calculations. The sides opposite the 45-degree angles, known as the legs, are congruent, meaning they have the same length. The side opposite the 90-degree angle is called the hypotenuse, which is the longest side of the triangle. A critical relationship in a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle is the ratio of the sides. If we denote the length of each leg as x, then the length of the hypotenuse is x√2. This ratio stems directly from the Pythagorean Theorem, which states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, this translates to x² + x² = (x√2)², which simplifies to 2x² = 2x², confirming the validity of the relationship. This consistent relationship is incredibly useful when solving problems where one side length is known, and the others need to be determined. By understanding and applying this ratio, we can efficiently find the missing side lengths without resorting to more complex trigonometric calculations. This makes the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle a cornerstone in various fields, from basic geometry to more advanced applications in physics and engineering.

In our problem, we are given that the hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 4 cm. We need to find the length of one leg of the triangle. Knowing the relationship between the legs and the hypotenuse in a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle is the key to solving this problem efficiently. As we established, if the length of each leg is x, then the length of the hypotenuse is x√2. In this case, the hypotenuse is given as 4 cm, so we can set up the equation x√2 = 4. To solve for x, which represents the length of one leg, we need to isolate x. This can be done by dividing both sides of the equation by √2. Thus, we have x = 4/√2. Now, to rationalize the denominator, we multiply both the numerator and the denominator by √2. This gives us x = (4√2) / (√2 * √2), which simplifies to x = (4√2) / 2. Finally, we simplify the fraction by dividing both the numerator and the denominator by 2, resulting in x = 2√2. Therefore, the length of one leg of the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle is 2√2 cm. This methodical approach, utilizing the specific properties of the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, allows us to quickly and accurately determine the unknown side length. Understanding these relationships not only helps in solving mathematical problems but also enhances our ability to visualize and analyze geometric shapes in various practical scenarios.

To find the length of one leg of the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle with a hypotenuse of 4 cm, we can follow these steps:

1. Recall the relationship: In a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the hypotenuse is √2 times the length of each leg.
2. Set up the equation: Let the length of one leg be x. Then, the hypotenuse is x√2. We are given that the hypotenuse is 4 cm, so we have x√2 = 4.
3. Solve for x: To find x, divide both sides of the equation by √2: x = 4/√2.
4. Rationalize the denominator: Multiply both the numerator and the denominator by √2: x = (4√2) / (√2 * √2) = (4√2) / 2.
5. Simplify: Divide both the numerator and the denominator by 2: x = 2√2.

Therefore, the length of one leg of the triangle is 2√2 cm. This step-by-step process clearly demonstrates how the properties of special right triangles can be used to solve geometric problems efficiently. By understanding these relationships, we can avoid more complex calculations and quickly arrive at the correct solution. The ability to work through these problems systematically is a valuable skill in mathematics and other quantitative fields, allowing for accurate and efficient problem-solving.

The length of one leg of the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle is B. $2 \&sqrt{2} ext{ cm}$.

In conclusion, understanding the properties of special right triangles, such as the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, is essential for solving geometric problems. The consistent relationships between the sides of these triangles allow us to quickly determine unknown lengths when given other information. In the case of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, the relationship that the hypotenuse is √2 times the length of each leg is particularly useful. By applying this knowledge, we were able to efficiently solve the problem where the hypotenuse was given as 4 cm, and we needed to find the length of one leg. The step-by-step solution demonstrated how to set up the equation, solve for the unknown, and rationalize the denominator to arrive at the final answer of 2√2 cm. This methodical approach not only provides the correct solution but also reinforces the importance of understanding the underlying principles of geometry. Mastery of these concepts can significantly enhance problem-solving skills in various mathematical contexts and real-world applications. By grasping the unique characteristics of special right triangles, we can approach geometric problems with confidence and precision, making complex calculations more manageable and intuitive.