Finding The Longest Side Of A 30-60-90 Triangle

Hey math enthusiasts! Today, we're diving into a geometry problem involving a special type of triangle: the 30-60-90 triangle. These triangles are super cool because their angles have specific measures ($30^{\circ}, 60^{\circ},$ and $90^{\circ}$), which leads to some neat relationships between their sides. We'll be using this knowledge to figure out the length of the longest side of a triangle, given its perimeter. So, buckle up, grab your pencils, and let's get started!

Understanding 30-60-90 Triangles: The Basics

First off, let's refresh our memory about 30-60-90 triangles. These right triangles have a unique property: their sides are always in a specific ratio. The sides are related to each other based on the angles. If you know the length of one side, you can figure out the lengths of the other two using these ratios. The side lengths follow this pattern: if the shortest side (opposite the 30-degree angle) has length x, then the hypotenuse (the longest side, opposite the 90-degree angle) has length 2x, and the remaining side (opposite the 60-degree angle) has length x√3. It's like a secret code for triangles, isn't it? Understanding this ratio is the key to solving our problem.

Now, let's talk about the perimeter. The perimeter of any shape, including a triangle, is simply the total length of all its sides added together. In our case, we're given that the perimeter of the 30-60-90 triangle is $18 + 6 ext{√}3$. Our goal is to use this information, along with our knowledge of the side ratios, to find the length of the longest side (the hypotenuse).

To begin, imagine we have a 30-60-90 triangle. Let's label the shortest side (opposite the 30-degree angle) as x. As we discussed, the side opposite the 60-degree angle will be x√3, and the hypotenuse (the longest side) will be 2x. The perimeter is the sum of all sides, which means we can express the perimeter in terms of x: $x + x ext{√}3 + 2x$. Now, let's set this expression equal to the given perimeter, $18 + 6 ext{√}3$, and solve for x. This will help us find the side lengths.

So, we have the equation: $x + x ext{√}3 + 2x = 18 + 6 ext{√}3$. Combining the x terms on the left side, we get: $(3 + ext{√}3)x = 18 + 6 ext{√}3$. This equation involves an expression with a square root, so we'll need to use some algebraic manipulation. The objective is to isolate x to determine the size of the sides.

Solving for the Sides: Step by Step

Alright guys, let's get down to the nitty-gritty and solve for x. Remember that equation we got? $(3 + ext{√}3)x = 18 + 6 ext{√}3$. To isolate x, we need to divide both sides of the equation by $(3 + ext{√}3)$. This gives us: x = $(18 + 6 ext{√}3) / (3 + ext{√}3)$. Now, it looks a bit messy with that square root in the denominator, so let's rationalize the denominator. This means we're going to multiply both the numerator and denominator by the conjugate of $(3 + ext{√}3)$, which is $(3 - ext{√}3)$.

So, we have:

x = $((18 + 6 ext{√}3) * (3 - ext{√}3)) / ((3 + ext{√}3) * (3 - ext{√}3))$.

Let's expand the numerator: $(18 * 3) + (18 * - ext{√}3) + (6 ext{√}3 * 3) + (6 ext{√}3 * - ext{√}3) = 54 - 18 ext{√}3 + 18 ext{√}3 - 18$. Simplifying this gives us: $36$.

Now, let's expand the denominator: $(3 * 3) + (3 * - ext{√}3) + ( ext{√}3 * 3) + ( ext{√}3 * - ext{√}3) = 9 - 3 ext{√}3 + 3 ext{√}3 - 3$. Simplifying this gives us: $6$.

So, our equation now looks like this: x = $36 / 6$. Therefore, x = $6$. We've found it, dudes! The length of the shortest side (opposite the 30-degree angle) is 6.

Now that we know x (the shortest side) is 6, we can easily find the lengths of the other two sides. The side opposite the 60-degree angle is x√3, which is $6 ext{√}3$. The hypotenuse (the longest side, opposite the 90-degree angle) is 2x, which is $2 * 6 = 12$.

Finding the Longest Side: The Grand Finale

We're almost there! We've successfully calculated the lengths of all three sides of our 30-60-90 triangle. The shortest side is 6, the side opposite the 60-degree angle is $6 ext{√}3$, and the hypotenuse (the longest side) is 12. Remember, the hypotenuse is always opposite the right angle (the 90-degree angle).

Therefore, the length of the longest side of the triangle is 12. Boom! We solved it! We successfully utilized the properties of a 30-60-90 triangle and our knowledge of perimeter calculations to determine the length of the hypotenuse. Wasn't that fun?

Just to recap, we started with the given perimeter of $18 + 6 ext{√}3$ and used the side ratios of a 30-60-90 triangle to set up an equation. After some algebraic manipulation (including rationalizing the denominator), we found the value of x. Using x, we then calculated the lengths of all sides, ultimately determining that the longest side (the hypotenuse) has a length of 12. Math can be so satisfying, right?

Key Takeaways and Further Exploration

Alright, let's summarize what we've learned and explore some additional ideas. The main takeaways from this problem are:

• Understanding 30-60-90 Triangle Ratios: Knowing the relationship between the sides (x, x√3, and 2x) is crucial for solving these types of problems.
• Perimeter Basics: The perimeter is simply the sum of all the sides of a shape.
• Algebraic Manipulation: We utilized algebraic techniques such as solving equations and rationalizing the denominator to solve for the unknown side lengths.

Now, for some further exploration: You could try similar problems with different perimeters. How would the solution change if the perimeter was given as a decimal or a more complex expression? Also, you might want to explore other special right triangles, such as the 45-45-90 triangle. These triangles have their own unique side ratios and problem-solving techniques.

Additionally, you can challenge yourself by tackling problems that combine concepts. For instance, you could be given the area of a 30-60-90 triangle and be asked to find its perimeter. This requires you to know how to calculate the area (Area = 0.5 * base * height) and utilize the side ratios in conjunction with the given area. Remember, the more you practice, the more comfortable you'll become with these concepts.

And finally, consider the real-world applications of these concepts. Right triangles and trigonometry are used in various fields, like architecture, engineering, and navigation. Understanding these foundational principles opens doors to a deeper appreciation for the role of mathematics in the world around us.

Keep practicing, keep exploring, and keep the math vibes going! You've got this, guys! And remember, if you have any questions or want to explore further math topics, just ask. The world of mathematics is vast and exciting, and there's always something new to learn and discover. So, keep your minds sharp and your calculators ready. See you next time, math adventurers!