Finding The Third Side Of A Triangle With Given Perimeter And Two Sides

Introduction

In geometry, understanding the relationships between the sides and perimeter of a triangle is fundamental. This article delves into a problem where we are given the lengths of two sides of a triangle in terms of a variable x, as well as the perimeter, also expressed in terms of x. Our goal is to find the length of the third side. This problem not only reinforces basic algebraic manipulation but also highlights the practical application of geometric principles. To solve this, we will use the basic formula for the perimeter of a triangle, which is the sum of the lengths of all its three sides.

Problem Statement

Consider a triangle where the lengths of two sides are (3x + 2) meters and (6x + 3) meters respectively. The perimeter of this triangle is given as (11x + 25/3) meters. The task is to determine the length of the third side of this triangle. This problem is a classic example of applying algebraic principles to geometric figures, specifically focusing on the properties of triangles. The perimeter of any polygon, including a triangle, is the sum of the lengths of all its sides. Therefore, to find the third side, we will subtract the sum of the two given sides from the total perimeter. This involves combining like terms and simplifying the algebraic expression to arrive at the length of the unknown side. This exercise not only tests algebraic skills but also enhances the understanding of geometric relationships, providing a solid foundation for more advanced topics in both algebra and geometry.

Setting up the Equation

To find the third side of the triangle, we start by expressing the perimeter in terms of the three sides. Let the length of the third side be y meters. According to the problem statement, the two given sides are (3x + 2) meters and (6x + 3) meters, and the perimeter is (11x + 25/3) meters. The perimeter of a triangle is the sum of the lengths of its three sides. Therefore, we can write the equation as follows:

Perimeter = Side 1 + Side 2 + Side 3

Substituting the given values, we get:

(11x + 25/3) = (3x + 2) + (6x + 3) + y

This equation forms the basis for solving the problem. The next step involves simplifying the equation by combining like terms. This algebraic manipulation will help isolate y, which represents the length of the third side. The process includes adding the expressions on the right side of the equation, which are the lengths of the two known sides, and then subtracting this sum from the perimeter. This is a standard algebraic technique used to solve for an unknown variable in an equation. By setting up the equation correctly and systematically simplifying it, we can accurately determine the length of the third side of the triangle. This problem demonstrates the practical application of algebraic equations in solving geometric problems, reinforcing the connection between these two branches of mathematics.

Solving for the Third Side

Now, let's solve the equation (11x + 25/3) = (3x + 2) + (6x + 3) + y for y. First, we combine the like terms on the right side of the equation. We add the x terms (3x + 6x), which gives us 9x, and then add the constants (2 + 3), which gives us 5. So, the equation becomes:

(11x + 25/3) = 9x + 5 + y

Next, we want to isolate y on one side of the equation. To do this, we subtract (9x + 5) from both sides of the equation. This gives us:

(11x + 25/3) - (9x + 5) = y

Now, we simplify the left side of the equation by distributing the negative sign and combining like terms. We subtract 9x from 11x, which gives us 2x. Then, we subtract 5 from 25/3. To do this, we need a common denominator, so we rewrite 5 as 15/3. Now we have:

25/3 - 15/3 = 10/3

So, the equation simplifies to:

2x + 10/3 = y

Therefore, the length of the third side, y, is (2x + 10/3) meters. This solution demonstrates the application of algebraic principles to solve geometric problems. By setting up the equation correctly and systematically simplifying it, we have successfully determined the length of the third side of the triangle in terms of x. This algebraic manipulation is a crucial skill in mathematics, allowing us to solve for unknown quantities in various contexts.

Verifying the Solution

To ensure the correctness of our solution, it is essential to verify it. We found that the third side of the triangle is (2x + 10/3) meters. Now, we will add the lengths of all three sides and check if the sum equals the given perimeter, (11x + 25/3) meters. The three sides are (3x + 2) meters, (6x + 3) meters, and (2x + 10/3) meters. Adding these together, we get:

(3x + 2) + (6x + 3) + (2x + 10/3)

First, let's combine the x terms: 3x + 6x + 2x = 11x.

Next, let's combine the constants: 2 + 3 + 10/3. To add these, we need a common denominator, so we rewrite 2 as 6/3 and 3 as 9/3. Now we have:

6/3 + 9/3 + 10/3 = 25/3

So, the sum of the lengths of the three sides is:

11x + 25/3

This sum is exactly the given perimeter of the triangle. Therefore, our solution for the length of the third side, (2x + 10/3) meters, is correct. This verification step is a crucial part of problem-solving in mathematics. It helps to catch any potential errors and ensures that the final answer is accurate and consistent with the given information. By verifying our solution, we gain confidence in our understanding of the problem and the methods used to solve it. This process also reinforces the importance of attention to detail and accuracy in mathematical calculations.

Conclusion

In conclusion, we have successfully found the length of the third side of the triangle by applying basic algebraic principles and geometric concepts. Given the lengths of two sides as (3x + 2) meters and (6x + 3) meters, and the perimeter as (11x + 25/3) meters, we determined the length of the third side to be (2x + 10/3) meters. This problem illustrates the practical application of algebra in solving geometric problems. By setting up the equation based on the perimeter formula, simplifying the expression, and solving for the unknown variable, we were able to find the length of the missing side. The verification step further confirmed the accuracy of our solution, reinforcing the importance of checking our work in mathematical problem-solving. This exercise not only enhances algebraic skills but also deepens the understanding of geometric relationships, providing a solid foundation for more advanced topics in both mathematics and related fields. The ability to translate geometric problems into algebraic equations and solve them is a valuable skill that has applications in various fields, including engineering, physics, and computer science. This problem serves as a good example of how mathematical concepts are interconnected and how they can be applied to solve real-world problems.