45-45-90 Triangle Problem Solving Hypotenuse 128 Cm

Understanding the properties of special right triangles, particularly the 45-45-90 triangle, is crucial in geometry and trigonometry. These triangles, with their consistent angle and side ratios, offer shortcuts for solving problems that would otherwise require more complex calculations. In this article, we will delve into a problem involving a 45-45-90 triangle where the hypotenuse is given, and we need to find the length of one of its legs. This exploration will not only help solve the specific problem but also reinforce the fundamental principles governing these special triangles. By grasping these concepts, you'll be better equipped to tackle a wide range of geometry problems with confidence and precision. Let's embark on this geometric journey and unravel the relationship between the hypotenuse and legs of a 45-45-90 triangle.

Problem Statement

The problem at hand states: The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 128 cm. What is the length of one leg of the triangle?

To solve this, we will explore the unique characteristics of 45-45-90 triangles and utilize the specific ratio that exists between their sides. These triangles, also known as isosceles right triangles, possess two equal angles of 45 degrees and one right angle of 90 degrees. This special configuration leads to a consistent relationship between the lengths of their sides, which we will leverage to find the unknown leg length. By understanding and applying this ratio, we can efficiently determine the solution without resorting to more complex trigonometric methods. The following sections will delve deeper into the properties of these triangles and demonstrate how to apply them to solve this problem effectively. This approach not only provides the answer but also enhances our understanding of geometric relationships.

Understanding 45-45-90 Triangles

A 45-45-90 triangle, also known as an isosceles right triangle, is a special type of right triangle characterized by its angles measuring 45 degrees, 45 degrees, and 90 degrees. This unique configuration gives rise to a consistent and predictable relationship between the lengths of its sides. The two legs of the triangle, opposite the 45-degree angles, are of equal length, making it an isosceles triangle. The side opposite the 90-degree angle is the hypotenuse, which is the longest side of the triangle.

The key to understanding 45-45-90 triangles lies in the ratio of their sides. If we denote the length of each leg as 'x', then the length of the hypotenuse is given by $x\sqrt{2}$. This ratio stems from the Pythagorean theorem ($a^2 + b^2 = c^2$), where 'a' and 'b' are the legs and 'c' is the hypotenuse. In a 45-45-90 triangle, since the legs are equal, the equation becomes $x^2 + x^2 = c^2$, which simplifies to $2x^2 = c^2$. Taking the square root of both sides yields $c = x\sqrt{2}$. This fundamental relationship allows us to easily determine the length of any side if we know the length of another side. For instance, if we know the length of a leg, we can find the hypotenuse by multiplying it by $\sqrt{2}$. Conversely, if we know the hypotenuse, we can find the length of a leg by dividing the hypotenuse by $\sqrt{2}$. This simple yet powerful ratio makes 45-45-90 triangles highly predictable and easy to work with in geometric problems. Understanding this ratio is essential for efficiently solving problems involving these special triangles.

Applying the 45-45-90 Triangle Ratio to the Problem

In our specific problem, we are given that the hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle measures 128 cm. Our goal is to find the length of one of the legs of the triangle. As established earlier, the relationship between the legs and the hypotenuse in a 45-45-90 triangle is such that if the length of each leg is 'x', then the hypotenuse has a length of $x\sqrt{2}$. Therefore, we can set up an equation to solve for 'x', the length of a leg.

Given that the hypotenuse is 128 cm, we can write the equation as: $x\sqrt{2} = 128$. To isolate 'x', we need to divide both sides of the equation by $\sqrt{2}$: $x = \frac{128}{\sqrt{2}}$. To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{2}$: $x = \frac{128\sqrt{2}}{2}$. Simplifying the fraction, we get: $x = 64\sqrt{2}$. Therefore, the length of one leg of the triangle is $64\sqrt{2}$ cm. This method demonstrates the direct application of the 45-45-90 triangle ratio to solve for an unknown side length. By understanding and utilizing this ratio, we can efficiently solve problems involving these special triangles without resorting to more complex trigonometric calculations. The solution clearly illustrates the power and convenience of recognizing and applying the properties of special right triangles in geometry.

Solution

Based on our calculations, the length of one leg of the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle is $64\sqrt{2}$ cm. This corresponds to option B in the given choices. To reiterate, we arrived at this solution by leveraging the fundamental relationship between the sides of a 45-45-90 triangle. We knew that the hypotenuse is $\sqrt{2}$ times the length of each leg. By setting up the equation $x\sqrt{2} = 128$, where 'x' represents the length of a leg, and solving for 'x', we found the leg length to be $64\sqrt{2}$ cm. This straightforward approach highlights the efficiency of understanding and applying the properties of special right triangles.

Conclusion

In conclusion, the length of one leg of the $45^{\circ}-45^{\circ}-90^{\circ}$ triangle with a hypotenuse of 128 cm is $64\sqrt{2}$ cm. This problem effectively demonstrates the importance of recognizing and applying the specific properties of special right triangles, such as the 45-45-90 triangle. The consistent ratio between the sides of these triangles allows for quick and efficient solutions to geometric problems. By understanding that the hypotenuse is $\sqrt{2}$ times the length of a leg in a 45-45-90 triangle, we can easily solve for unknown side lengths. This knowledge not only simplifies problem-solving but also deepens our understanding of geometric relationships. Mastering these concepts is crucial for success in geometry and related fields, providing a solid foundation for tackling more complex problems. The ability to recognize and apply these principles showcases a strong grasp of geometric fundamentals.