Triangle Height Calculation: Step-by-Step Guide

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Hey guys! Let's dive into a super practical math problem today: finding the height of a triangle when we know its area and base. This is something that comes up a lot in geometry, and it’s really useful to understand. We'll break it down step by step, so don't worry if it seems tricky at first. By the end of this, you'll be a pro at using the formula A = (1/2)bh!

Understanding the Basics

Before we jump into the problem, let’s make sure we’re all on the same page with the basic concepts. So, when dealing with triangles, a solid grasp of the formula for its area is absolutely essential. The area of a triangle is calculated using the formula A = (1/2)bh, where:

  • A represents the area of the triangle.
  • b stands for the base of the triangle, which is one of its sides.
  • h denotes the height of the triangle, which is the perpendicular distance from the base to the opposite vertex (the highest point).

Imagine a triangle sitting on a flat surface. The base is the side resting on the surface, and the height is how tall the triangle is from that surface to its tip. This height is super important because it tells us how much space the triangle covers. Think of it like this: if you were to fill the triangle with paint, the area tells you how much paint you’d need. So, if we know the area and the length of the base, we can work backward to figure out the height. It’s like solving a puzzle, where we have some pieces and need to find the missing one. Got it? Great! Now, let’s move on to the problem we have at hand. We're given the area and the base, and our mission is to find that height using the formula we just talked about. Stick with me, and we'll solve it together!

Problem Statement

Let's clearly state the problem. We have a triangle with the following information:

  • Area (A) = 29.7 cm²
  • Base (b) = 13.5 cm

Our mission, should we choose to accept it (and we do!), is to find the height (h) of this triangle. We’ll use the formula A = (1/2)bh to accomplish this. We know the values for A and b, and we need to rearrange the formula to solve for h. It’s like having a recipe where we know the total amount of cookies we want and how much flour we used, but we need to figure out how much sugar to add. We'll use the same kind of thinking here. So, the key is to manipulate the formula so that h is all by itself on one side of the equation. This might sound intimidating, but don’t worry! We’ll take it step by step, and you'll see it’s not as scary as it seems. In the next section, we’ll dive into the actual math and show you exactly how to do this. Ready to get started? Let’s go!

Applying the Formula

Alright, let's get down to business! We're going to use the formula A = (1/2)bh to find the height of our triangle. The first thing we need to do is plug in the values we know. We know the area (A) is 29.7 cm² and the base (b) is 13.5 cm. So, let's substitute those values into the formula:

  1. 7 = (1/2) * 13.5 * h

Now we have an equation with one unknown, h. Our goal is to isolate h on one side of the equation. To do this, we need to get rid of the other numbers around h. The first step is to simplify the right side of the equation. We have (1/2) multiplied by 13.5. What’s half of 13.5? It's 6.75. So, we can rewrite the equation as:

  1. 7 = 6.75 * h

Now, h is being multiplied by 6.75. To get h by itself, we need to do the opposite operation, which is division. We'll divide both sides of the equation by 6.75. Remember, whatever we do to one side of the equation, we have to do to the other side to keep it balanced. This is like making sure both sides of a scale have the same weight so it stays even. So, let’s divide:

  1. 7 / 6.75 = (6.75 * h) / 6.75

This simplifies to:

  1. 4 = h

So, we’ve found that the height (h) is 4.4 cm. Awesome! We’ve successfully used the formula to solve for the height. In the next section, we'll double-check our answer to make sure it makes sense. Keep going, you're doing great!

Solution

So, after plugging in the values and doing a little bit of algebra magic, we found that the height (h) of the triangle is 4.4 cm. That’s our solution! But before we do a victory dance, let's just take a moment to double-check our work. It's always a good idea to make sure our answer makes sense in the context of the problem. We can do this by plugging our calculated height back into the original formula and seeing if we get the area we started with.

Let's use the formula A = (1/2)bh again. This time, we know A (29.7 cm²) and b (13.5 cm), and we think we know h (4.4 cm). Let's plug those values in:

Area = (1/2) * 13.5 cm * 4.4 cm

Now, let’s calculate:

Area = (1/2) * 59.4 cm² Area = 29.7 cm²

Ta-da! The area we calculated matches the area we were given in the problem. This means our height of 4.4 cm is correct. We’ve not only found the solution, but we’ve also verified it. That’s how you nail a math problem! So, the correct answer is:

D. h = 4.4 cm

Nice work, guys! You’ve successfully found the height of the triangle. In the next section, we'll wrap things up and recap the steps we took.

Conclusion

Alright, we’ve reached the end of our triangle adventure! Let's take a moment to recap what we've done. We started with a problem: finding the height of a triangle given its area and base. We knew the area was 29.7 cm² and the base was 13.5 cm. We had a trusty tool in our toolbox – the formula for the area of a triangle, A = (1/2)bh. This formula is your best friend when dealing with triangles, so make sure you keep it handy!

We then plugged the values we knew into the formula. This gave us an equation with the height (h) as the only unknown. From there, it was all about algebra. We simplified the equation, isolated h, and found that the height was 4.4 cm. But we didn’t stop there! We double-checked our answer by plugging the height back into the formula to make sure we got the correct area. This is a super important step in problem-solving. Always verify your solution if you can. It’s like proofreading an essay before you turn it in. You want to catch any mistakes before they cost you points!

So, what have we learned? We’ve learned how to use the area formula to find the height of a triangle. This is a valuable skill that you can use in all sorts of math problems. Remember, the key is to understand the formula, plug in the values you know, and then use your algebra skills to solve for the unknown. And most importantly, don’t forget to check your work! You guys did an awesome job today. Keep practicing, and you'll become math superstars in no time!