Solving For Isosceles Triangle Sides A Step By Step Guide

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Hey guys! Let's dive into a geometry problem together. We've got an interesting one here involving an isosceles triangle, and we're going to break it down step-by-step so it's super clear. Our main goal? To figure out the lengths of all the sides. So, grab your thinking caps, and let's get started!

Understanding the Problem: Isosceles Triangles and Perimeter

In this geometry problem, isosceles triangles are the stars of the show. Remember, an isosceles triangle is a triangle with two sides that are equal in length. These equal sides are often called the legs, while the third side is the base. The problem gives us a crucial piece of information: the side of our isosceles triangle (which we'll assume to be one of the legs) is one and a half times longer than its base. This relationship between the sides is key to solving the puzzle. The perimeter of a triangle is simply the sum of the lengths of all its sides. In our case, we know the perimeter is 44 cm. This is our total length constraint, and it’s going to help us form our equation. Combining these two pieces of information – the relationship between the sides and the total perimeter – we can set up an equation to find the actual lengths. Think of it like this: we have a triangle where two sides are linked in length, and we know the total length of all three sides combined. This allows us to use algebra to “backtrack” and find the individual side lengths. It's like having a recipe where you know the total amount of ingredients and the ratio of some ingredients, and you need to figure out the exact amount of each ingredient. Before we jump into the equation, it's always a good idea to visualize the problem. Imagine an isosceles triangle. Picture the two equal sides being 1.5 times longer than the base. This visual representation can make the algebraic steps feel more intuitive. In the next section, we'll translate this visual and conceptual understanding into a concrete algebraic equation. We'll use variables to represent the unknown side lengths and then use the given information to link these variables together. This is where the problem transforms from a geometric puzzle into an algebraic one, which we can then solve using familiar techniques.

Setting Up the Equation: Translating Geometry into Algebra

Alright, let's get down to the nitty-gritty and turn this geometry problem into an algebra equation! The first step is to assign variables to the unknowns. Since the base of the triangle is the foundation for the other sides, let's call the length of the base "x". This means one of our unknowns is now represented algebraically. Remember, the problem states that the side of the isosceles triangle (the leg) is one and a half times the length of the base. So, how do we represent that algebraically? Well, 1.5 times x is simply 1.5x. Since we have two equal sides in an isosceles triangle, both legs will have a length of 1.5x. Now we have algebraic representations for all three sides: the base is x, and each of the two equal sides is 1.5x. This is a crucial step because it allows us to use the power of algebra to solve for the unknown lengths. Next, we bring in the information about the perimeter. We know the perimeter is the sum of all the sides, and we know the perimeter is 44 cm. So, we can write an equation that represents this: x (the base) + 1.5x (one leg) + 1.5x (the other leg) = 44. This equation is the heart of the solution. It captures all the information given in the problem in a concise algebraic form. It links the unknown side length (x) to the known perimeter (44 cm). Now, before we start solving, let’s take a moment to appreciate what we’ve done. We've taken a geometric description and turned it into a mathematical statement. This is a powerful skill in problem-solving! The equation we've created is a linear equation, which means it's relatively straightforward to solve. In the next step, we'll simplify the equation by combining like terms. This will make the equation even easier to work with and bring us closer to finding the value of x, which is the length of the base. Once we have the length of the base, we can easily find the lengths of the other sides since they are defined in terms of x.

Solving the Equation: Finding the Value of x

Now comes the fun part – actually solving the equation! We've got our equation: x + 1.5x + 1.5x = 44. The first step is to simplify the equation by combining like terms. On the left side, we have three terms that all involve x. So, we can add their coefficients together: 1x + 1.5x + 1.5x. If you add those up, you get 4x. So, our simplified equation is now 4x = 44. See how much cleaner that looks? Combining like terms makes the equation easier to grasp and reduces the chance of making errors in the next steps. Now, we need to isolate x to find its value. Remember, the goal of solving an equation is to get the variable by itself on one side of the equals sign. In this case, x is being multiplied by 4. To undo this multiplication, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 4. This is a crucial step to maintain the balance of the equation. What we do to one side, we must do to the other. So, we have (4x) / 4 = 44 / 4. On the left side, the 4s cancel out, leaving us with just x. On the right side, 44 divided by 4 is 11. Therefore, we have x = 11. Boom! We've found the value of x. But what does this mean in the context of our triangle? Remember, x represents the length of the base of the isosceles triangle. So, we now know the base is 11 cm long. We're not quite done yet, though. We still need to find the lengths of the other two sides. But don't worry, we're in the home stretch. In the next section, we'll use the value of x to calculate the lengths of the legs of the triangle and complete our solution. It's like we've found a key piece of the puzzle, and now we can use it to unlock the rest.

Finding the Sides: Completing the Solution

Okay, we've cracked the code and found that x = 11 cm, which is the length of the base of our isosceles triangle. But we're not quite at the finish line yet! We still need to find the lengths of the other two sides, the legs of the triangle. Remember, the problem told us that each leg is 1.5 times the length of the base. And we now know the base is 11 cm. So, to find the length of a leg, we simply multiply 1.5 by 11. Let's do the math: 1. 5 * 11 = 16.5. So, each leg of the isosceles triangle is 16.5 cm long. Fantastic! We've found the lengths of all three sides: the base is 11 cm, and each leg is 16.5 cm. But before we declare victory, it's always a good idea to double-check our work. A great way to do this is to make sure our solution fits the original problem statement. The problem told us the perimeter of the triangle is 44 cm. So, let's add up the lengths of the sides we found and see if they equal 44 cm: 11 cm (base) + 16.5 cm (leg) + 16.5 cm (leg). If you add those up, you get 44 cm! This confirms that our solution is correct. We've successfully found the lengths of all three sides of the isosceles triangle and verified that they add up to the given perimeter. We took a geometric problem, translated it into an algebraic equation, solved the equation, and then used the solution to find the side lengths. This is a classic example of how algebra and geometry work together to solve problems. So, to summarize our solution, the sides of the isosceles triangle are 11 cm, 16.5 cm, and 16.5 cm. Great job, everyone! We tackled this problem together and came out on top.

Final Answer

Therefore, the sides of the triangle are:

  • Base: 11 cm
  • Sides: 16.5 cm each