Equilateral Triangle Height: Solved!

Hey math enthusiasts! Let's dive into a geometry problem that's super common. We're going to figure out the height of an equilateral triangle, specifically triangle FGH. We know its sides are $34 \sqrt{3}$ units long. We'll break down the question, go through the solution step-by-step, and explain why everything works. It is going to be fun, and you'll become a pro at these triangle problems! So, buckle up!

Understanding the Problem: The Equilateral Triangle

Alright, let's start with the basics. The question presents us with an equilateral triangle, which is awesome because it means all three sides are equal in length and all three angles are equal as well, each measuring 60 degrees. Knowing this is a total game-changer, guys. The problem tells us that each side of triangle FGH is $34\sqrt{3}$ units long. The question wants us to find the height of the triangle. The height of a triangle is a line that goes from one corner (vertex) straight down to the opposite side, forming a perfect right angle (90 degrees). In an equilateral triangle, the height also splits the base into two equal parts, making it a perpendicular bisector. This means we can use the Pythagorean theorem to solve it. Let's look at the options given to us. We need to decide which among the options A. 17 units, B. 34 units, C. 51 units, or D. 68 units is the correct height of the triangle. Seems simple, right? It totally is. Knowing this helps us visualize the problem and set up the right way to solve it. Get ready to do some simple math!

This basic understanding is crucial. The question itself is pretty straightforward, but recognizing the special properties of an equilateral triangle is the key to solving it efficiently. The goal here isn't just to get the right answer; it's to grasp why the solution works. That way, you can tackle similar problems with confidence. The ability to visualize the shape and its properties makes the math a breeze. Always remember to draw a picture if it helps! That is a pro tip. So, the first step is always to understand the terms. Then we can go on and start calculating our answer.

Solving for the Height: Step-by-Step

Now, let's get down to the nitty-gritty and find the height. Here's how we'll do it. Since the height of an equilateral triangle cuts the base exactly in half, we're essentially dealing with two right-angled triangles inside our main triangle. Let's call the point where the height meets the base point M. This creates two right triangles: FGM and HGM. The base of each of these smaller triangles is half the length of the original side. Given that the entire side is $34\sqrt{3}$ units, the base of each small triangle (either FM or HM) is $\frac{34\sqrt{3}}{2} = 17\sqrt{3}$ units. Got it? Next, the side of the original triangle, which is also the hypotenuse of the right-angled triangle, is $34\sqrt{3}$ units. We need to find the height, which we'll call 'h'. The Pythagorean theorem states: $a^2 + b^2 = c^2$, where 'a' and 'b' are the shorter sides of the right-angled triangle, and 'c' is the longest side (the hypotenuse). In our case, $a = 17\sqrt{3}$, $c = 34\sqrt{3}$, and 'b' is the height 'h' we are looking for. So, let's put the values in the equation: $(17\sqrt{3})^2 + h^2 = (34\sqrt{3})^2$. When we simplify this: $(17^2 * 3) + h^2 = (34^2 * 3)$. Then we simplify further: $867 + h^2 = 3468$. To find 'h', subtract 867 from both sides to get $h^2 = 2601$. Now, we need to take the square root of both sides to get 'h', which is $h = \sqrt{2601} = 51$ units. So the height of the triangle FGH is 51 units. Not so hard, right? This is an easy way of finding the answer, and now we will go over some key points.

The Pythagorean theorem is super helpful here. That's why understanding this is so important, because it allows us to easily find the missing side when we already know the other two sides. The math may seem complicated at first, but with a bit of practice, you'll get the hang of it and solve these kinds of problems in no time. Always remember to break down the problem into smaller, more manageable steps. It makes the whole process less overwhelming. Don't worry if it takes a bit of time at first. Practice makes perfect, and you'll soon be able to solve these problems like a pro! The key is to visualize the problem, understand the theorem, and carefully apply the formula. Easy peasy, right?

The Answer and Explanation

So, after all that calculation, we found that the height of triangle FGH is 51 units. That means the correct answer is option C. We found the correct answer by using the properties of an equilateral triangle and applying the Pythagorean theorem. Because all sides of an equilateral triangle are equal, knowing the side length allows us to calculate the height. The height, in this case, divides the triangle into two congruent right-angled triangles. By understanding these geometrical properties and applying a well-known formula, the solution becomes quite straightforward. It all boils down to knowing the basic rules and formulas and how they apply in different scenarios. Also, understanding the theorem is critical because it's a fundamental concept in geometry that helps us solve all sorts of problems related to triangles. Knowing this means we have a solid grasp of how triangles work. Congratulations on finding the correct answer!

Let's recap what we did: we started with an equilateral triangle, understood its properties, broke it down into smaller right-angled triangles, and then applied the Pythagorean theorem to find the height. Pretty neat, huh? Next time you see a problem like this, you'll know exactly what to do. Always remember the properties of different types of triangles – it helps a lot. It is going to give you a big advantage in geometry, and you'll be able to solve similar problems with ease. And now you know how to find the height of an equilateral triangle! Now go and tell your friends!