Triangle Perimeter Calculation: Step-by-Step Solution

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Calculating the Perimeter of Triangle AED: A Step-by-Step Guide

Hey guys! Let's dive into a geometry problem where we need to figure out the perimeter of a triangle. We've got a rectangle, some side lengths, and a point hanging out on one of the sides. Sounds like a fun puzzle, right? So, let's break it down and solve it together!

Understanding the Problem: The Rectangle and the Triangle

First things first, let's visualize what we're dealing with. We have a rectangle, let's call it ABCD. Now, rectangles have some cool properties: their opposite sides are equal in length, and all their angles are right angles (that's 90 degrees for those keeping score). We know that side AD is 25 cm and side AB is 12 cm. Since it's a rectangle, we also know that BC is equal to AD (25 cm) and CD is equal to AB (12 cm). Got it? Great!

Now, here's where it gets a tad more interesting. There's a point, E, chilling on side BC. We're told that the length of BE is 9 cm. This is crucial information because it helps us figure out the length of the remaining part of side BC, which is EC. To find EC, we simply subtract BE from BC: EC = BC - BE = 25 cm - 9 cm = 16 cm. So, now we know EC is 16 cm. We're slowly piecing together the puzzle, aren't we?

Our mission, should we choose to accept it (and we do!), is to calculate the perimeter of triangle AED. Remember, the perimeter of any shape is just the sum of the lengths of all its sides. So, for triangle AED, we need to find the lengths of AE, ED, and DA. We already know DA is 25 cm (it's the same as AD, a side of the rectangle). So, we're halfway there, in terms of sides, at least!

Let's recap: We have a rectangle ABCD, AD = 25 cm, AB = 12 cm, a point E on BC such that BE = 9 cm, and we need to find the perimeter of triangle AED. We've figured out that EC = 16 cm and DA = 25 cm. Now, we need to find AE and ED. Time to roll up our sleeves and do some more math!

Calculating the Missing Sides: AE and ED

Okay, so we've got two sides of our triangle left to figure out: AE and ED. To do this, we're going to use a trusty tool from our mathematical toolkit: the Pythagorean theorem. Remember that one? It's the one that says 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 simpler terms: a² + b² = c², where c is the hypotenuse.

First, let's tackle AE. If you look closely at our rectangle, you'll notice that triangle ABE is a right-angled triangle (because all angles in a rectangle are right angles). The right angle is at B, so AE is the hypotenuse. We know AB is 12 cm and BE is 9 cm. So, we can plug these values into the Pythagorean theorem:

AE² = AB² + BE² AE² = 12² + 9² AE² = 144 + 81 AE² = 225

To find AE, we need to take the square root of 225. The square root of 225 is 15. So, AE = 15 cm. Awesome! One side down, one to go.

Now, let's find ED. Similarly, triangle ECD is also a right-angled triangle, with the right angle at C. ED is the hypotenuse in this case. We know CD is 12 cm (same as AB) and we figured out earlier that EC is 16 cm. Let's use the Pythagorean theorem again:

ED² = EC² + CD² ED² = 16² + 12² ED² = 256 + 144 ED² = 400

To find ED, we take the square root of 400. The square root of 400 is 20. So, ED = 20 cm. Fantastic! We've found the last side.

We've successfully navigated through the triangles and used the Pythagorean theorem to uncover the lengths of AE and ED. AE is 15 cm, and ED is 20 cm. Now, we're just one step away from our final answer. Let's bring it home!

Calculating the Perimeter: Putting It All Together

Alright, guys, we're in the home stretch! We've done the heavy lifting of figuring out the lengths of all the sides of triangle AED. Now, it's just a simple matter of adding them up to find the perimeter. Remember, the perimeter is the total distance around the shape.

We know:

  • AE = 15 cm
  • ED = 20 cm
  • DA = 25 cm

So, the perimeter of triangle AED is:

Perimeter = AE + ED + DA Perimeter = 15 cm + 20 cm + 25 cm Perimeter = 60 cm

And there you have it! The perimeter of triangle AED is 60 cm. We did it! We took a geometry problem, broke it down into smaller, manageable parts, and used our knowledge of rectangles, right-angled triangles, and the Pythagorean theorem to find the solution.

Final Answer and Key Takeaways

The perimeter of triangle AED is 60 cm. High fives all around!

But more than just getting the right answer, let's think about what we learned along the way. This problem wasn't just about plugging numbers into formulas. It was about:

  • Visualizing the problem: Drawing a diagram helps immensely in understanding the relationships between the different elements.
  • Breaking down complex problems: We divided the big problem into smaller, easier-to-solve parts.
  • Using the right tools: The Pythagorean theorem was our key tool in finding the missing side lengths.
  • Step-by-step approach: We followed a logical sequence of steps to arrive at the solution.

So, the next time you encounter a geometry problem (or any problem, really), remember these strategies. They'll help you tackle even the trickiest challenges. Keep practicing, keep learning, and keep having fun with math! You've got this!