Calculating Pyramid Height: An Equilateral Triangle Guide

Hey math enthusiasts! Let's dive into a geometry problem that's all about finding the height of a triangular base of a solid oblique pyramid. The base of our pyramid is an equilateral triangle with a base edge length of 18 inches. We're going to break down how to find the height, and I promise, it's not as scary as it sounds! Let's get started.

Understanding the Problem: The Equilateral Triangle

Alright, first things first, let's make sure we're all on the same page. We're dealing with an equilateral triangle, which is a triangle where all three sides are equal in length. In our case, each side of the triangle is 18 inches long. The question wants us to find the height of this triangle. Remember, the height of a triangle is the perpendicular distance from the base to the opposite vertex (the point where the other two sides meet). This height splits the equilateral triangle into two 30-60-90 right triangles.

To find the height, we're going to use some handy-dandy geometry and a little bit of the Pythagorean theorem, which is super useful for right triangles. This theorem tells us that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). So, let's draw a picture and see how it works.

Imagine our equilateral triangle. Now, draw a line from one corner (vertex) straight down to the middle of the opposite side. This line is our height, and it creates two right triangles. The base of each of these right triangles is half the length of the original side of the equilateral triangle (because the height bisects the base). The hypotenuse of each right triangle is one of the original sides of the equilateral triangle. If you draw it out, it really helps visualize everything! In our case, the base of each little right triangle is 9 inches (18 inches / 2), and the hypotenuse is 18 inches.

Now, we'll apply the Pythagorean theorem: $a^2 + b^2 = c^2$, where 'c' is the hypotenuse, and 'a' and 'b' are the other two sides. We know 'c' (18 inches) and one of the legs, which is half of the original side (9 inches). We need to find the height, which is the other leg ('b'). So, we can rearrange the theorem to solve for 'b'.

Solving for the Height

Let's get down to the nitty-gritty and calculate the height. We have our right triangle with a hypotenuse of 18 inches and one leg of 9 inches. Using the Pythagorean theorem, we have: $a^2 + b^2 = c^2$. Plugging in our values: $9^2 + b^2 = 18^2$. That simplifies to $81 + b^2 = 324$. Now, subtract 81 from both sides to isolate $b^2$: $b^2 = 324 - 81$, which gives us $b^2 = 243$. To find 'b' (the height), we need to take the square root of 243. So, $b = \sqrt{243}$.

Here’s a little trick for simplifying square roots: we can break down 243 into its prime factors. 243 = 3 * 81. And 81 is 9 * 9, or 9^2. So, we can rewrite $\sqrt{243}$ as $\sqrt{3 * 81}$, which simplifies to $\sqrt{3 * 9^2}$. When you take the square root of a number squared, you just get the original number, so $\sqrt{9^2}$ is just 9. That leaves us with $9\sqrt{3}$.

So, the height (b) of the equilateral triangle is $9\sqrt{3}$ inches! That's our answer. Let's look back at the options you provided and see which one matches.

Identifying the Correct Answer Choice

Okay, we've done the math, and we have our answer: the height of the equilateral triangle base is $9\sqrt{3}$ inches. Now, let's take a look at the answer choices you gave and see which one matches our result. Remember, we calculated the height using the Pythagorean theorem, working with the sides of the equilateral triangle and breaking it down into right triangles.

Here are the options again:

A. $9 \sqrt{2}$ in. B. $9 \sqrt{3}$ in. C. $18 \sqrt{2}$ in. D. $18 \sqrt{3}$ in.

Looking at our options, we can clearly see that option B. $9 \sqrt{3}$ in. is the correct answer. This is the value we calculated, so we can confidently choose it. Awesome! You've successfully found the height of the triangular base of the pyramid. Great job!

This problem highlights how understanding basic geometry principles, like the properties of an equilateral triangle and the Pythagorean theorem, can help you solve more complex problems. These concepts are fundamental in various areas of mathematics and are useful in real-world applications. Keep practicing, and you'll become a geometry whiz in no time!

Conclusion: Geometry is Awesome!

We did it, guys! We successfully calculated the height of the triangular base of the pyramid. We used our knowledge of equilateral triangles, the Pythagorean theorem, and a little bit of square root simplification to arrive at the correct answer. Remember, the key is to break down complex problems into smaller, more manageable steps. By understanding the properties of shapes and applying the right formulas, you can solve almost any geometry problem.

Keep practicing, and don't be afraid to draw diagrams to help visualize the problem. Geometry can be super fun when you break it down into manageable parts. And if you ever get stuck, don’t hesitate to ask for help or look for examples. Math is all about learning and growing, so keep up the great work. You've got this!

I hope this step-by-step guide was helpful. If you have any more questions or want to try another problem, feel free to ask! Geometry is a fascinating field, and with a little practice, you can master it. Congrats on finding the height of that triangle – you’re doing great! Keep up the awesome work, and keep exploring the amazing world of mathematics! You're well on your way to becoming a geometry master. Keep practicing, keep learning, and keep having fun with it! You've got the skills to tackle any problem that comes your way. Way to go!