Scalene Triangle Perimeter Problem Solving Equations And Side Lengths

In the realm of geometry, triangles hold a fundamental position, and among them, scalene triangles stand out with their unique characteristics. A scalene triangle, by definition, is a triangle with all three sides of different lengths. This distinctive property leads to varying angles as well, making it an intriguing subject of study. To fully grasp scalene triangles, it's essential to understand the concept of perimeter, which is the total distance around the outside of any shape. For a triangle, the perimeter is simply the sum of the lengths of its three sides. This article delves into a specific problem involving a scalene triangle, where the perimeter and the relationship between the sides are given, and we aim to determine the equation that can be used to find the side lengths.

Problem Statement

Consider a scalene triangle with a perimeter of 14.5 cm. We are given that the longest side is twice the length of the shortest side. Furthermore, the longest side measures 6.2 cm. The challenge here is to identify the correct equation that can be used to calculate the lengths of the sides of this triangle. This problem not only tests our understanding of geometric principles but also our ability to translate word problems into algebraic equations, a crucial skill in mathematical problem-solving.

Breaking Down the Problem

To approach this problem effectively, we need to break it down into smaller, manageable parts. First, let's define the key terms and variables: perimeter, scalene triangle, longest side, and shortest side. Understanding these terms is fundamental to constructing the correct equation. Next, we need to identify the relationships between the sides as described in the problem statement. The problem explicitly states that the longest side is twice the length of the shortest side. This is a critical piece of information that will help us form our equation. Lastly, we will use the given information about the perimeter to set up an equation that we can solve for the unknown side lengths. By systematically analyzing the problem, we can transform it into an algebraic expression that accurately represents the given conditions.

Setting Up the Equation

Now, let's translate the problem into an algebraic equation. We know the perimeter of the triangle is 14.5 cm, and the longest side is 6.2 cm. Let's denote the shortest side as 'a' cm. According to the problem, the longest side is twice the shortest side, which means 6.2 cm = 2a. From this, we can find the length of the shortest side. The remaining side, which we'll call 'b', is unknown. The perimeter, which is the sum of all three sides, can be expressed as 6.2 + a + b = 14.5. The question asks for the equation that can be used to find the side lengths, and among the choices, $6.2+b=14.5$ seems closest, but it's crucial to understand what 'b' represents in this context. It's not just any side; it's the remaining side after accounting for the longest side. To correctly set up the equation, we need to incorporate the relationship between the shortest side and the longest side. This step is vital in ensuring that the equation accurately reflects the geometric properties of the scalene triangle and the given conditions.

Analyzing the Options

The given option is:

A. $6.2+b=14.5$

To determine if this is the correct equation, let's analyze what each term represents in the context of the problem. The number 6.2 represents the length of the longest side, which is given. The variable 'b' represents the length of the remaining side, which is neither the longest nor the shortest. The sum of these two sides, 6.2 + b, is equated to 14.5, which is the perimeter of the triangle. However, this equation seems to be missing a crucial component: the shortest side. The perimeter of a triangle is the sum of all three sides, not just two. Therefore, the equation must include the length of the shortest side as well. Let's denote the shortest side as 'a'. We know that the longest side (6.2 cm) is twice the shortest side, so we can write 6.2 = 2a. Solving for 'a', we get a = 3.1 cm. Now, we can express the perimeter as the sum of all three sides: 6.2 + 3.1 + b = 14.5. Simplifying this equation, we get 9.3 + b = 14.5. This is different from the given option, which suggests that the given equation is incomplete. The correct equation should account for all three sides of the triangle and their relationships.

Why Option A Might Be Misleading

Option A, $6.2 + b = 14.5$, is a partially correct equation but doesn't fully represent the problem's conditions. While it acknowledges the longest side (6.2 cm) and the remaining side (b), it omits the shortest side. This omission is significant because the shortest side plays a crucial role in defining the triangle and its perimeter. The problem statement explicitly mentions the relationship between the longest and shortest sides, which is a key piece of information that must be incorporated into the equation. By excluding the shortest side, the equation fails to capture the complete picture of the triangle's dimensions. This highlights the importance of carefully considering all the information provided in a problem statement and ensuring that the equation accurately reflects all the given conditions. A seemingly correct equation can be misleading if it doesn't account for all the variables and relationships involved.

Constructing the Correct Equation

To construct the correct equation, we need to incorporate all the information provided in the problem statement. We know the perimeter is 14.5 cm, the longest side is 6.2 cm, and the longest side is twice the shortest side. Let's denote the shortest side as 'a' cm and the remaining side as 'b' cm. From the given information, we can write the following equations:

1. Longest side = 2 * Shortest side, which translates to 6.2 = 2a.
2. Perimeter = Longest side + Shortest side + Remaining side, which translates to 14.5 = 6.2 + a + b.

From the first equation, we can solve for 'a':

a = 6.2 / 2 = 3.1 cm.

Now, we substitute the value of 'a' into the second equation:

14. 5 = 6.2 + 3.1 + b.

Simplifying the equation, we get:

15. 5 = 9.3 + b.

To isolate 'b', we subtract 9.3 from both sides:

b = 14.5 - 9.3 = 5.2 cm.

So, the lengths of the sides are 6.2 cm, 3.1 cm, and 5.2 cm. While we have found the lengths of the sides, the question asks for the equation that can be used to find these lengths. The equation we derived from the perimeter is 14.5 = 6.2 + a + b. However, since we know a = 3.1 cm, we can rewrite the equation as 14.5 = 6.2 + 3.1 + b, which simplifies to 14.5 = 9.3 + b. This is the equation that accurately represents the problem and can be used to find the length of the remaining side. Comparing this to the given option, we can see that option A, $6.2 + b = 14.5$, is not the complete equation. The correct equation should include the shortest side as well.

The Importance of Accurate Equations

The process of constructing and analyzing equations is fundamental to solving mathematical problems, especially in geometry. An accurate equation is essential because it serves as a mathematical representation of the problem, capturing the relationships between the variables and the given conditions. A flawed equation, on the other hand, can lead to incorrect solutions and a misunderstanding of the underlying concepts. In this case, the equation $6.2 + b = 14.5$ is not entirely accurate because it omits the shortest side of the triangle. The correct equation must account for all three sides to accurately represent the perimeter. This highlights the importance of carefully analyzing the problem statement, identifying all the relevant information, and translating it into an equation that fully captures the problem's essence. A well-constructed equation is the foundation for a correct solution, and it also demonstrates a clear understanding of the mathematical principles involved.

Conclusion

In conclusion, the problem of finding the side lengths of a scalene triangle with a given perimeter and a relationship between the longest and shortest sides requires a careful analysis of the given information and the construction of an accurate equation. The option $6.2 + b = 14.5$ is not the correct equation because it does not include the shortest side. The correct approach involves identifying all three sides and their relationships, setting up the perimeter equation, and solving for the unknown side lengths. This problem demonstrates the importance of understanding geometric principles, translating word problems into algebraic expressions, and ensuring that equations accurately represent the given conditions. By mastering these skills, we can confidently tackle a wide range of mathematical problems and deepen our understanding of the world around us.

This exercise not only reinforces our understanding of scalene triangles and perimeter calculations but also highlights the importance of precision and attention to detail in mathematical problem-solving. The ability to construct accurate equations and interpret them correctly is a valuable skill that extends beyond the classroom and into various real-world applications.