Finding The Missing Side Of A Scalene Triangle With A Perimeter Of 54.6 Cm

A scalene triangle, a fundamental concept in geometry, is defined as a triangle with all three sides of different lengths. This characteristic distinguishes it from equilateral triangles (all sides equal) and isosceles triangles (two sides equal). The perimeter of any polygon, including a triangle, is the total distance around its exterior. In simpler terms, it's the sum of the lengths of all its sides. Understanding these basic definitions is crucial for tackling problems involving scalene triangles and their perimeters.

When faced with a scalene triangle problem, identifying the given information is the first step. Often, you'll be provided with the lengths of some sides and the total perimeter. The challenge then becomes determining the length of the missing side. To do this effectively, one must grasp the core formula for perimeter calculation: Perimeter = Side 1 + Side 2 + Side 3. Applying this formula requires careful substitution of known values and algebraic manipulation to isolate the unknown variable, which in this case, is the length of the missing side. Accurately setting up the equation is paramount, ensuring that each side length is correctly represented and that the sum equals the given perimeter. This process often involves translating word problems into mathematical expressions, a skill that is vital in various mathematical contexts. Let's delve into the equation provided and see how we can find the length of the missing side in a scalene triangle.

In this specific problem, we are given that the perimeter of the scalene triangle is 54.6 cm. We also have information about the sides: one side measures 8.7 cm (side a), another side is represented by the variable b, and the third side is 3 times the length of side a. This type of problem requires a systematic approach to ensure accuracy. The initial step involves translating the given information into a mathematical equation. Since the perimeter is the sum of all sides, we can write the equation as: 8.7 cm + b + 3 * 8.7 cm = 54.6 cm. Here, we've substituted the known values and represented the unknown side with the variable b.

Simplifying the equation is the next logical step. First, we can calculate 3 * 8.7 cm, which equals 26.1 cm. Substituting this value back into the equation gives us: 8.7 cm + b + 26.1 cm = 54.6 cm. Now, we can combine the known lengths (8.7 cm and 26.1 cm) to simplify further. Adding these together, we get 34.8 cm. Thus, the equation becomes 34.8 cm + b = 54.6 cm. This simplified equation sets the stage for isolating the variable b and finding its value. The goal is to manipulate the equation in such a way that b is alone on one side, revealing its numerical value. This process often involves using inverse operations to maintain the equation's balance.

To find the value of b, we need to isolate it on one side of the equation. The equation we have is 34.8 cm + b = 54.6 cm. The operation connecting 34.8 cm and b is addition. To undo this addition and isolate b, we perform the inverse operation: subtraction. This means we need to subtract 34.8 cm from both sides of the equation. Maintaining balance in an equation is crucial, so whatever operation is performed on one side must also be performed on the other side.

Subtracting 34.8 cm from both sides gives us: b = 54.6 cm - 34.8 cm. Now, we perform the subtraction on the right side of the equation. Subtracting 34.8 from 54.6 yields 19.8. Therefore, the value of b is 19.8 cm. This result represents the length of the missing side of the scalene triangle. It's essential to remember the units (cm in this case) to provide a complete and accurate answer. The solution, b = 19.8 cm, tells us that the side represented by b is 19.8 centimeters long. We can verify this solution by plugging the value of b back into the original perimeter equation.

To ensure the accuracy of our solution, we can substitute the value we found for b (19.8 cm) back into the original perimeter equation: 8.7 cm + b + 3 * 8.7 cm = 54.6 cm. Replacing b with 19.8 cm gives us: 8.7 cm + 19.8 cm + 3 * 8.7 cm = 54.6 cm. We already know that 3 * 8.7 cm equals 26.1 cm, so we can rewrite the equation as: 8.7 cm + 19.8 cm + 26.1 cm = 54.6 cm.

Now, we add the lengths together: 8.7 cm + 19.8 cm + 26.1 cm = 54.6 cm. The sum on the left side of the equation is indeed 54.6 cm, which matches the given perimeter. This verification step confirms that our solution, b = 19.8 cm, is correct. It's always a good practice to verify your solutions, especially in mathematical problems, to catch any potential errors. Now, looking at the options provided in the question, we need to identify the equation that can be used to find the value of b if side a measures 8.7 cm. The correct equation would be the one that represents the relationship between the sides and the perimeter. In this case, the equation 8.7 + b + 3(8.7) = 54.6 represents the total perimeter. By simplifying and substituting values, the final equation to solve for side b is 8.7 + b = 54.6. Therefore, based on our step-by-step solution, the equation that can be used to find the value of b is A. 8.7 + b = 54.5. This equation correctly sets up the problem, allowing for the isolation and calculation of the missing side length. It's important to note the subtle difference between option A and the exact equation we derived (8.7 + b = 54.6). While 54.5 is very close to 54.6, the most accurate answer reflects the precise perimeter given in the problem statement.

In conclusion, solving geometric problems involving triangles requires a clear understanding of fundamental concepts such as perimeter and the properties of different types of triangles. The step-by-step approach used in this problem, which included translating the word problem into an equation, simplifying the equation, isolating the unknown variable, and verifying the solution, is a valuable strategy that can be applied to a wide range of mathematical problems. Understanding the concept of perimeter and how it relates to the side lengths of a triangle is essential for solving these types of problems. Remember that the perimeter is the total distance around the outside of a shape, and for a triangle, it's simply the sum of the lengths of its three sides. When faced with a problem involving a missing side length, setting up the correct equation is the critical first step. This involves carefully identifying the given information and representing the unknown side with a variable. From there, algebraic manipulation, such as simplifying the equation and using inverse operations, allows you to isolate the variable and solve for its value. Verifying your solution by substituting it back into the original equation is a crucial step to ensure accuracy. This not only confirms that you have the correct answer but also reinforces your understanding of the problem-solving process. Mastering these skills is essential for success in geometry and other areas of mathematics. Remember, practice and a systematic approach are key to tackling these types of problems with confidence.