Solving For Triangle Side Lengths Using Perimeter An Easy Guide
In the fascinating world of geometry, triangles hold a special place. These three-sided polygons come in various shapes and sizes, each with unique properties and characteristics. One common problem in geometry involves determining the side lengths of a triangle given its perimeter and some relationships between the sides. This article delves into such a problem, providing a step-by-step solution and exploring the underlying mathematical concepts. Understanding the dimensions of a triangle is crucial not only in academic settings but also in practical applications such as engineering, architecture, and design. Triangles are fundamental geometric shapes, and mastering their properties is essential for problem-solving in various fields. This article aims to demystify the process of finding the side lengths of a triangle when the perimeter and relationships between sides are known. We will explore the step-by-step solution to the problem, making it easier for you to understand and apply these concepts in different scenarios.
Problem Statement Unveiling the Triangle's Mystery
Let's consider a specific problem to illustrate the process. Suppose we have a triangle where the base measures (x - 2) cm. The first sloping side is twice the length of the base, and the second sloping side is three times the length of the base. The perimeter of this triangle is given as 90 cm. Our goal is to find the length of each side of the triangle. The perimeter of a triangle is the sum of the lengths of all its sides. In this case, we know the perimeter is 90 cm, and we have expressions for the lengths of each side in terms of a variable, x. By setting up an equation, we can solve for x and then find the actual lengths of the sides. Understanding how to translate word problems into mathematical equations is a key skill in algebra and geometry. This problem provides an excellent opportunity to practice this skill and reinforce your understanding of triangle properties. The challenge lies in correctly interpreting the given information and formulating the equation that represents the perimeter of the triangle.
Setting Up the Equation Transforming Words into Math
The first step in solving this problem is to translate the given information into a mathematical equation. We know that the base of the triangle is (x - 2) cm. The first sloping side is twice the base, which can be expressed as 2 * (x - 2) cm. The second sloping side is three times the base, or 3 * (x - 2) cm. Since the perimeter is the sum of the lengths of all sides, we can write the equation as follows:
(x - 2) + 2(x - 2) + 3(x - 2) = 90
This equation represents the perimeter of the triangle in terms of x. The next step is to simplify and solve for x. Setting up the equation correctly is crucial for arriving at the correct solution. Each term in the equation represents the length of one side of the triangle, and their sum equals the given perimeter. By carefully translating the word problem into a mathematical equation, we have laid the foundation for solving for the unknown variable.
Solving for x Unlocking the Variable's Value
Now that we have the equation, we need to solve for x. First, let's simplify the equation by distributing the constants and combining like terms:
x - 2 + 2x - 4 + 3x - 6 = 90
Combine the x terms and the constant terms:
6x - 12 = 90
Next, add 12 to both sides of the equation:
6x = 102
Finally, divide both sides by 6 to solve for x:
x = 17
So, the value of x is 17. This means that we can now find the lengths of each side of the triangle. Solving for x is a fundamental algebraic skill, and in this case, it allows us to determine the numerical value of the variable that represents a key dimension of the triangle. The steps involved in solving for x, including distribution, combining like terms, and isolating the variable, are essential techniques in algebra. By correctly applying these techniques, we have found the value of x, which is a crucial step in determining the lengths of the triangle's sides.
Calculating Side Lengths Putting x to Work
With the value of x determined, we can now calculate the length of each side of the triangle. Recall that the base is (x - 2) cm. Substituting x = 17, we get:
Base = 17 - 2 = 15 cm
The first sloping side is 2(x - 2) cm. Substituting x = 17, we get:
First Sloping Side = 2 * (17 - 2) = 2 * 15 = 30 cm
The second sloping side is 3(x - 2) cm. Substituting x = 17, we get:
Second Sloping Side = 3 * (17 - 2) = 3 * 15 = 45 cm
Therefore, the lengths of the sides of the triangle are 15 cm, 30 cm, and 45 cm. Calculating the side lengths involves substituting the value of x back into the expressions for each side. This step demonstrates how algebraic solutions can be used to find concrete values in geometric problems. By correctly substituting x and performing the arithmetic, we have successfully determined the lengths of the three sides of the triangle.
Verification Ensuring the Solution's Accuracy
To ensure our solution is correct, we can verify that the sum of the side lengths equals the given perimeter. The lengths we found are 15 cm, 30 cm, and 45 cm. Let's add them together:
15 + 30 + 45 = 90 cm
Since the sum of the side lengths is equal to the given perimeter of 90 cm, our solution is correct. Verification is an important step in problem-solving, as it confirms the accuracy of the solution and helps catch any potential errors. By verifying that the sum of the side lengths matches the given perimeter, we can be confident in our solution.
Conclusion Mastering Triangle Problems
In this article, we successfully found the lengths of the sides of a triangle given its perimeter and relationships between the sides. We translated the word problem into a mathematical equation, solved for the unknown variable, and calculated the side lengths. This problem demonstrates the application of algebraic principles in solving geometric problems. Mastering triangle problems requires a solid understanding of algebraic techniques and geometric concepts. By breaking down the problem into smaller steps, such as setting up the equation, solving for the variable, and calculating the side lengths, we can effectively tackle these types of problems. This process not only enhances our mathematical skills but also improves our problem-solving abilities in general. The key takeaways from this article include the importance of translating word problems into mathematical equations, the use of algebraic techniques to solve for unknowns, and the application of these solutions to geometric contexts. By practicing these skills, you can confidently approach similar problems and deepen your understanding of mathematics.
This detailed walkthrough not only provides the solution to the specific problem but also explains the underlying concepts and techniques. This approach can help readers better understand the process and apply it to other similar problems. The combination of a clear explanation, step-by-step solution, and verification makes this article a valuable resource for anyone looking to improve their problem-solving skills in geometry.
