Triangle Area: Calculation With Perimeter 24a

Hey everyone! Today, we're diving into a fun geometry problem where we need to figure out the area of a triangle. We're given some cool info, like the perimeter and the lengths of two sides. So, let's put on our math hats and get started!

Understanding the Triangle Basics

Before we jump into the calculations, let's quickly recap some triangle basics. A triangle, as you guys know, is a shape with three sides and three angles. The area of a triangle tells us the amount of space it covers, kind of like how much carpet you'd need to cover the floor. The perimeter, on the other hand, is the total distance around the triangle – imagine walking along all three sides; that's the perimeter!

The Area Formula: A = (b * h) / 2

Now, the key to finding the area is this neat little formula: A = (b * h) / 2. Here, 'A' stands for the area, 'b' is the length of the base of the triangle, and 'h' is the height. The base is simply one of the sides of the triangle, and the height is the perpendicular distance from the base to the opposite corner (the vertex). Think of it like drawing a straight line from the top of the triangle down to the base, making a perfect right angle.

Perimeter: The Sum of All Sides

Remember, the perimeter is the total length of all the sides added together. If we have a triangle with sides of lengths a, b, and c, then the perimeter (P) is: P = a + b + c. This simple formula will be super helpful in solving our problem.

Solving the Triangle Problem

Okay, let's tackle the problem at hand. We're given that the perimeter of our triangle is 24a, and two of its sides are each 8a long. Our mission is to find the area of this triangle. Sounds like a puzzle, right? Let's break it down step by step.

Step 1: Finding the Missing Side

First things first, we need to figure out the length of the third side. We know the perimeter is the sum of all sides, so we can write the equation:

24a = 8a + 8a + c

Where 'c' is the length of the missing side. Let's simplify this:

24a = 16a + c

To isolate 'c', we subtract 16a from both sides:

c = 24a - 16a

c = 8a

Great! We've found that the third side is also 8a long. This tells us something cool: our triangle is actually an isosceles triangle (a triangle with two equal sides) and, in fact, an equilateral triangle (a triangle with all three sides equal).

Step 2: Determining the Height

Now that we know all three sides are 8a, we need to find the height to use our area formula. Here's where things get a little more interesting. Since it's an equilateral triangle, we can draw a line from one vertex straight down to the middle of the opposite side (the base). This line is our height, and it also splits the equilateral triangle into two identical right-angled triangles.

To find the height, we can use the Pythagorean theorem, which states that in a right-angled 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 our case:

(8a)^2 = (4a)^2 + h^2

Why 4a? Because the height divides the base (8a) into two equal parts.

Let's simplify:

64a^2 = 16a^2 + h^2

Subtract 16a^2 from both sides:

48a^2 = h^2

Now, take the square root of both sides:

h = √(48a^2)

h = √(16 * 3 * a^2)

h = 4a√3

So, the height of our triangle is 4a√3.

Step 3: Calculating the Area

We're almost there! We have the base (8a) and the height (4a√3), so we can finally use the area formula:

A = (b * h) / 2

A = (8a * 4a√3) / 2

A = (32a^2√3) / 2

A = 16a^2√3

And there you have it! The area of the triangle is 16a^2√3. Awesome job, guys!

Key Takeaways

Let's quickly recap what we've learned in this mathematical adventure:

• Area of a Triangle: The area of a triangle is calculated using the formula A = (b * h) / 2, where 'b' is the base and 'h' is the height.
• Perimeter of a Triangle: The perimeter is the sum of all the sides of the triangle.
• Equilateral Triangles: Equilateral triangles have all three sides equal, which makes finding the height a bit easier using the Pythagorean theorem.
• Pythagorean Theorem: This theorem (a^2 + b^2 = c^2) is super useful for finding missing sides in right-angled triangles.

By understanding these concepts, you can tackle all sorts of triangle problems. Keep practicing, and you'll become a geometry whiz in no time!

Practice Problems

Want to test your skills? Try solving these similar problems:

1. A triangle has a perimeter of 30b. Two sides are 10b and 10b. Find the area.
2. An equilateral triangle has sides of length 6c. What is its area?

Share your answers in the comments below! Let's learn together!

Conclusion

So, guys, we've successfully calculated the area of a triangle given its perimeter and two side lengths. Remember, the key is to break down the problem into smaller, manageable steps. By understanding the formulas and applying the Pythagorean theorem, we can conquer even the trickiest geometry challenges. Keep exploring, keep learning, and most importantly, have fun with math!