Right Triangle Hypotenuse Length: A Step-by-Step Solution

Let's dive into a classic geometry problem! We're given a right triangle, a shortest side length of 3√3 inches, and one angle measuring 60°. Our mission, should we choose to accept it, is to find the length of the hypotenuse. Fear not, geometry enthusiasts, for we shall conquer this challenge with a blend of trigonometric knowledge and a dash of problem-solving prowess!

Understanding the Right Triangle

First, let's visualize what we're dealing with. A right triangle, as the name suggests, has one angle that's exactly 90 degrees. The side opposite the right angle is called the hypotenuse, and it's always the longest side. The other two sides are called legs. Since we're given that one angle is 60°, we can deduce that the third angle must be 30° (because the angles in any triangle always add up to 180°). This makes our triangle a special type of right triangle known as a 30-60-90 triangle.

Now, the key piece of information is that 3√3 inches is the length of the shortest side. In a 30-60-90 triangle, the shortest side is always opposite the smallest angle, which in this case is the 30° angle. This relationship is super important because it helps us connect the side lengths using trigonometric ratios or special triangle properties. Remember, trigonometry gives us the tools to relate angles and side lengths in triangles. Knowing the relationships between the sides in a 30-60-90 triangle allows for a quicker solution, but understanding trigonometry provides a more fundamental approach applicable to a wider range of problems. Understanding the properties of special right triangles, such as the 30-60-90 triangle, streamlines problem-solving and allows for quick and efficient calculations. These triangles possess specific side length ratios that, once memorized, can be directly applied to find missing lengths without resorting to trigonometric functions each time. Specifically, in a 30-60-90 triangle, the side lengths are in the ratio of 1:√3:2, where the shortest side (opposite the 30° angle) corresponds to 1, the longer leg (opposite the 60° angle) corresponds to √3, and the hypotenuse corresponds to 2. This ratio holds true regardless of the actual side lengths, making it a powerful tool for solving problems involving these special triangles. For instance, if you know the length of the shorter leg, you can easily find the lengths of the other two sides by simply multiplying by the appropriate factor. Similarly, if you know the hypotenuse, you can divide by 2 to find the shorter leg and then multiply by √3 to find the longer leg. This direct application of ratios eliminates the need for complex calculations and saves valuable time, especially during exams. While trigonometric functions provide a universal approach to solving triangle problems, mastering the properties of special right triangles offers a shortcut that can significantly enhance your problem-solving speed and accuracy. Therefore, it's highly recommended to familiarize yourself with these special triangles and their associated side length ratios.

Using Trigonometry to Find the Hypotenuse

Let's bring in our trigonometric functions! We'll use the sine function. Recall that the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In our case, we have:

sin(60°) = (opposite side) / (hypotenuse)

We know the angle (60°) and the length of the side opposite the other angle, which is 3√3 inches. We want to find the hypotenuse, let's call it h. So, we can write:

sin(60°) = 3√3 / h

Now, we need to remember the value of sin(60°). If you have it memorized, great! If not, you can quickly sketch a 30-60-90 triangle and use the side ratios to figure it out. sin(60°) is √3 / 2. Plugging that in, we get:

√3 / 2 = 3√3 / h

To solve for h, we can cross-multiply:

h√3 = 2 * 3√3

h√3 = 6√3

Now, divide both sides by √3:

h = 6

Therefore, the length of the hypotenuse is 6 inches.

Trigonometric functions, such as sine, cosine, and tangent, provide a powerful toolset for relating angles and side lengths in right triangles. These functions are based on the ratios of the sides of a right triangle with respect to its angles. Specifically, the sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse, the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse, and the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. By utilizing these functions, we can solve for unknown side lengths or angles in a right triangle if we know at least one side length and one angle (other than the right angle). For instance, in the problem at hand, we were given the length of the shortest side (opposite the 30° angle) and the measure of one of the acute angles (60°). Using the sine function, we could relate the length of the opposite side to the length of the hypotenuse and solve for the unknown hypotenuse. The trigonometric approach offers a versatile and systematic method for tackling a wide range of right triangle problems, regardless of whether the triangle is a special right triangle or not. Moreover, understanding the underlying principles of trigonometry provides a solid foundation for tackling more advanced concepts in mathematics and physics.

Alternative Approach: Using 30-60-90 Triangle Properties

Since we know this is a 30-60-90 triangle, we can use its special properties to solve the problem even faster! In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. The shortest side (opposite the 30° angle) corresponds to 1, the side opposite the 60° angle corresponds to √3, and the hypotenuse corresponds to 2.

We're given that the side opposite the 60° angle (the √3 side) has a length of 3√3 inches. This means that the "√3" in our ratio actually represents 3√3 inches. To find what "1" represents (the side opposite the 30° angle), we can divide 3√3 by √3, which gives us 3. So, the side opposite the 30° angle is 3 inches.

Now, to find the hypotenuse, which corresponds to "2" in our ratio, we simply multiply the length of the shortest side (3 inches) by 2. This gives us 6 inches.

Voila! The hypotenuse is 6 inches. This method is quicker if you remember the 30-60-90 triangle ratios, but the trigonometric approach is more fundamental and applicable to any right triangle.

Understanding the unique properties of special right triangles, such as the 30-60-90 triangle, offers a powerful shortcut for solving geometry problems. These triangles possess specific side length ratios that, once memorized, can significantly expedite calculations. In a 30-60-90 triangle, the side lengths are always in the ratio of 1:√3:2, where the shortest side (opposite the 30° angle) corresponds to 1, the longer leg (opposite the 60° angle) corresponds to √3, and the hypotenuse corresponds to 2. This ratio remains constant regardless of the actual side lengths, allowing for direct application in solving problems. For instance, if you know the length of the shortest side, you can easily find the lengths of the other two sides by simply multiplying by the appropriate factor. Conversely, if you know the length of the hypotenuse, you can divide by 2 to find the length of the shortest side and then multiply by √3 to find the length of the longer leg. Mastering these ratios not only saves time but also provides a deeper understanding of the relationships between the sides of a 30-60-90 triangle. Furthermore, recognizing and applying these properties can simplify complex problems, making them more manageable and less prone to errors. Therefore, it's highly recommended to familiarize yourself with the special right triangles and their associated side length ratios to enhance your problem-solving skills in geometry.

Conclusion

We've successfully navigated the right triangle terrain and found the length of the hypotenuse to be 6 inches! Whether you prefer the trigonometric route or the 30-60-90 triangle shortcut, understanding the underlying principles is key to conquering these geometric challenges. Keep practicing, and you'll become a triangle-tackling titan in no time!

For further exploration of right triangles and trigonometry, check out resources like Khan Academy's Trigonometry Section.