Triangle Side Lengths Calculation Using Perimeter And Base Relationships
In this article, we'll dive into a geometric problem that involves finding the lengths of the sides of a triangle when we know its perimeter and the relationships between the sides. This is a classic problem in mathematics that combines algebra and geometry, providing a great example of how these two branches of mathematics intersect. We'll break down the problem step-by-step, making it easy to understand even if you're not a math whiz. So, let's get started!
Problem Statement
The problem we're tackling is this: The base of a triangle is (x - 2) cm. The first sloping side is twice the length of the base, and the second sloping side is 3 times the length of the base. If the perimeter of the triangle is 90 cm, our goal is to find the length of each side. To effectively address this problem, we will delve into a detailed, step-by-step solution, ensuring every aspect is clearly elucidated for better understanding. This approach not only helps in solving the immediate problem but also equips you with strategies applicable to similar mathematical challenges. The importance of understanding the fundamental relationship between a triangle's sides and its perimeter cannot be overstated, as it forms the bedrock for more advanced geometrical concepts. Perimeter, in its essence, is the total distance around a two-dimensional shape, which, in the case of a triangle, is simply the sum of the lengths of its three sides. This seemingly straightforward concept is pivotal in various mathematical and real-world applications, from calculating the fencing required for a garden to designing structures with specific dimensional constraints. Furthermore, the algebraic representation of side lengths, as given in our problem, is a common technique in mathematics that allows us to solve for unknown quantities using equations. The beauty of algebra lies in its ability to abstract real-world scenarios into symbolic expressions, which can then be manipulated to find solutions. This interplay between geometry and algebra is a recurring theme in mathematical problem-solving, and mastering it is crucial for anyone looking to deepen their understanding of mathematics. By working through this problem, we not only find the solution but also reinforce our grasp of these foundational concepts, setting a solid base for tackling more complex problems in the future. Therefore, our approach will emphasize not just the 'how' of solving the problem but also the 'why,' ensuring a comprehensive learning experience.
Setting up the Equations
To solve this problem effectively, the first critical step is to translate the given information into mathematical equations. This process involves representing the unknown side lengths in terms of a variable and then using the perimeter to form an equation. Let's break down each part of the problem and express it algebraically. The base of the triangle is given as (x - 2) cm. This expression represents the length of one side of the triangle in terms of the variable 'x.' It's important to note that 'x' is an unknown quantity that we aim to find. The first sloping side is described as being twice the length of the base. Mathematically, this can be expressed as 2 * (x - 2) cm. This means that we take the length of the base and multiply it by 2 to get the length of the first sloping side. Similarly, the second sloping side is 3 times the length of the base, which translates to 3 * (x - 2) cm. This indicates that the length of the base is multiplied by 3 to find the length of the second sloping side. Now, we come to the key piece of information: the perimeter of the triangle is 90 cm. The perimeter, by definition, is the sum of the lengths of all the sides. Therefore, we can write an equation that represents this relationship: (x - 2) + 2(x - 2) + 3(x - 2) = 90. This equation is the cornerstone of our solution. It mathematically connects the variable 'x' to the known perimeter of the triangle. Solving this equation will give us the value of 'x,' which we can then use to find the lengths of each side. The process of setting up this equation is a perfect example of how algebra can be used to model real-world scenarios. By translating the word problem into an algebraic equation, we've created a tool that we can use to solve for the unknowns. This is a fundamental skill in mathematics and is applicable in a wide range of problem-solving situations. Therefore, understanding how to translate verbal information into mathematical expressions and equations is a crucial step in mastering problem-solving techniques.
Solving for x
With the equation (x - 2) + 2(x - 2) + 3(x - 2) = 90 established, the next vital step is to solve for the unknown variable, x. This involves simplifying the equation and isolating x on one side. Let's methodically break down the process. First, we need to simplify the equation by combining like terms. Notice that each term on the left side of the equation has a common factor of (x - 2). We can factor this out to make the simplification process clearer. The equation can be rewritten as (x - 2)(1 + 2 + 3) = 90. Now, we simplify the terms inside the parentheses: (x - 2)(6) = 90. This step significantly reduces the complexity of the equation, making it easier to work with. Next, we want to isolate the term (x - 2). To do this, we divide both sides of the equation by 6: (x - 2) = 90 / 6. Performing the division, we get (x - 2) = 15. Now, we have a much simpler equation. To solve for x, we need to get x by itself. We can do this by adding 2 to both sides of the equation: x = 15 + 2. Finally, we perform the addition to find the value of x: x = 17. So, we've successfully solved for x! This means we now know the numerical value that satisfies our equation. However, it's important to remember what x represents in the context of our problem. It's not the final answer but a crucial stepping stone towards finding the lengths of the sides of the triangle. The process of solving for x demonstrates the power of algebraic manipulation. By following a series of logical steps, we were able to untangle the equation and find the value of the unknown variable. This is a fundamental skill in mathematics and is used extensively in various fields, including science, engineering, and economics. The ability to solve equations is not just about finding the right answer; it's about developing a systematic approach to problem-solving, which is a valuable skill in any aspect of life.
Finding the Side Lengths
Now that we've determined the value of x to be 17, the crucial next step is to use this value to calculate the actual lengths of each side of the triangle. This involves substituting x = 17 back into the expressions we derived for the base, the first sloping side, and the second sloping side. Let's start with the base. We know that the base is given by the expression (x - 2) cm. Substituting x = 17, we get: Base = (17 - 2) cm = 15 cm. So, the length of the base of the triangle is 15 cm. Next, let's find the length of the first sloping side. This side is given by 2(x - 2) cm. Again, substituting x = 17, we have: First sloping side = 2(17 - 2) cm = 2(15) cm = 30 cm. Therefore, the first sloping side is 30 cm long. Finally, we need to calculate the length of the second sloping side. This side is given by 3(x - 2) cm. Substituting x = 17, we get: Second sloping side = 3(17 - 2) cm = 3(15) cm = 45 cm. Thus, the second sloping side measures 45 cm. We have now successfully found the lengths of all three sides of the triangle: the base is 15 cm, the first sloping side is 30 cm, and the second sloping side is 45 cm. It's always a good practice to check our solution to ensure it makes sense in the context of the problem. One way to do this is to add up the side lengths and see if they equal the given perimeter. Perimeter = Base + First sloping side + Second sloping side = 15 cm + 30 cm + 45 cm = 90 cm. Since the sum of the side lengths matches the given perimeter of 90 cm, we can be confident that our solution is correct. This process of finding the side lengths highlights the importance of careful substitution and calculation. It also underscores the value of checking our work to ensure accuracy. In mathematical problem-solving, precision is key, and every step, from setting up the equations to finding the final answer, should be carried out with diligence and attention to detail. Moreover, this exercise demonstrates how algebraic solutions translate back into concrete values in a geometric context, reinforcing the connection between these two branches of mathematics.
Conclusion
In conclusion, we have successfully found the lengths of the sides of the triangle using the given information about its perimeter and the relationships between its sides. The base of the triangle is 15 cm, the first sloping side is 30 cm, and the second sloping side is 45 cm. This problem provides a great example of how algebraic techniques can be applied to solve geometric problems. By translating the word problem into mathematical equations, we were able to find the unknown side lengths. This is a fundamental skill in mathematics and is applicable in a wide range of contexts. The key takeaways from this problem include the importance of carefully setting up equations, simplifying expressions, and solving for unknowns. It also highlights the value of checking our work to ensure accuracy. Mathematics is not just about finding the right answer; it's about developing a logical and systematic approach to problem-solving. This skill is valuable not only in academic settings but also in everyday life. Whether you're calculating the area of a room, figuring out a budget, or solving a complex problem at work, the principles of mathematical problem-solving can help you break down the problem into manageable steps and find a solution. Therefore, practicing problems like this is not just about getting a grade in math class; it's about developing skills that will serve you well in many areas of your life. The interplay between geometry and algebra, as demonstrated in this problem, is a recurring theme in mathematics. Many geometric problems can be solved using algebraic techniques, and vice versa. Understanding this connection is crucial for a deep understanding of mathematics. By mastering these fundamental concepts and problem-solving techniques, you'll be well-equipped to tackle more complex mathematical challenges in the future. Remember, mathematics is a journey, not a destination. Every problem you solve is a step forward on that journey, and the skills you develop along the way will empower you to tackle whatever challenges come your way.
