Finding The Height Of An Equilateral Triangle Base In An Oblique Pyramid

Hey guys! Let's dive into a geometry problem where we're dealing with an oblique pyramid. Specifically, we're looking at a pyramid with an equilateral triangle as its base. The base edge length is given as 18 inches, and our mission is to find the height of this triangular base. Sounds like fun, right? Let's break it down and make sure we understand every step.

Understanding the Equilateral Triangle

So, first things first, an equilateral triangle is a triangle where all three sides are equal in length. Not just that, but all three angles are also equal, each measuring 60 degrees. This is crucial because it gives us a lot of information to work with. In our case, each side of the triangle is 18 inches. When we talk about the height of this triangle, we're referring to the perpendicular distance from one vertex (corner) to the opposite side (the base). Think of it as the altitude of the triangle.

To find this height, we can use a little trick: split the equilateral triangle into two congruent right-angled triangles. Imagine drawing a line from one vertex straight down to the midpoint of the opposite side. This line is our height, and it neatly divides the equilateral triangle into two identical right triangles. Each of these right triangles has a hypotenuse (the longest side) that is 18 inches (the original side of the equilateral triangle). One of the legs (the shorter sides) is half the base of the equilateral triangle, which is 18 inches / 2 = 9 inches. And the other leg? That’s the height we’re trying to find! This is where the magic of geometry happens.

Applying the Pythagorean Theorem

Now that we've got ourselves a right triangle, we can bring in our trusty friend, the Pythagorean Theorem. Remember that? It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In math terms, it's ${a^2 + b^2 = c^2}$, where ${c}$ is the hypotenuse, and ${a}$ and ${b}$ are the legs. We know the hypotenuse is 18 inches, and one leg is 9 inches. Let's call the height ${h}$. So, we can write the equation as:

${ 9^2 + h^2 = 18^2 }$

Let's break this down step by step. First, calculate the squares:

${ 81 + h^2 = 324 }$

Next, we want to isolate ${h^2}$, so we subtract 81 from both sides of the equation:

${ h^2 = 324 - 81 }$

${ h^2 = 243 }$

Now, to find ${h}$, we need to take the square root of both sides:

${ h = \sqrt{243} }$

But wait, we're not done yet! We need to simplify that square root. Think of factors of 243 that are perfect squares. We can rewrite 243 as ${81 \times 3}$. So:

${ h = \sqrt{81 \times 3} }$

${ h = \sqrt{81} \times \sqrt{3} }$

Since the square root of 81 is 9, we get:

${ h = 9 \sqrt{3} }$

The Final Answer and Why It Makes Sense

So, the height of the triangular base of the pyramid is ${9 \sqrt{3}}$ inches. That's option B. Boom! We nailed it. But let’s just take a moment to think about why this answer makes sense. We started with an equilateral triangle, split it into right triangles, used the Pythagorean Theorem, and simplified our result. Each step was logical, and the final answer fits the geometry of the problem. It’s always a good idea to double-check your work and make sure the answer is reasonable in the context of the question.

Why the Other Options Aren't Correct

Let's quickly glance at the other options to see why they don't fit:

• A. ${9 \sqrt{2}}$ in.: This value is close, but it would imply a different relationship between the sides of the right triangle. The ${\sqrt{2}}$ typically shows up in 45-45-90 triangles, not 30-60-90 triangles (which is what we have here).
• C. ${18 \sqrt{2}}$ in.: This is double the value of option A and doesn't align with the proportions we've established.
• D. ${18 \sqrt{3}}$ in.: This would be the height if we hadn't divided the base in half when we formed the right triangle. It's a common mistake to overlook that step.

Key Takeaways for Solving Similar Problems

Alright, guys, let's wrap this up with some key takeaways. When you're faced with a geometry problem like this, remember these steps:

1. Understand the shapes: Know the properties of equilateral triangles, right triangles, and how they relate to each other.
2. Draw diagrams: Sketching the problem helps you visualize the relationships and break it down into smaller parts.
3. Apply the Pythagorean Theorem: This is a fundamental tool for solving right triangle problems.
4. Simplify radicals: Don’t leave your answer as a messy square root. Simplify it to its simplest form.
5. Check your work: Make sure your answer makes sense in the context of the problem.

Geometry can seem tricky at first, but with practice and a solid understanding of the basics, you'll be solving these problems like a pro. Keep up the great work, and I'll see you in the next problem!

Final Answer: The final answer is $\boxed{9 \sqrt{3} in}$