Perimeter Of A Triangle Is 16 Cm, One Side Is 5.5 Cm, And The Other Two Sides Are In The Ratio 3:4. Find The Lengths Of The Other Two Sides.

In geometry, triangles hold a fundamental place, and understanding their properties is crucial for various applications. One such property is the perimeter, which is the total length of all sides of the triangle. In this article, we delve into a problem where we are given the perimeter of a triangle and the ratio of its two sides, and our objective is to determine the lengths of those sides. This problem exemplifies how algebraic principles can be applied to solve geometric problems effectively. Let's embark on this mathematical journey and unravel the solution.

The question involves a triangle whose perimeter is given as 16 cm. We also know that one of its sides measures 5.5 cm, and the other two sides are in the ratio of 3:4. The challenge here is to find the exact lengths of these remaining two sides. To solve this, we will employ a blend of basic geometry and algebraic techniques. First, we will define the unknown sides using a variable, considering their given ratio. Then, we will set up an equation using the perimeter formula, which states that the perimeter of a triangle is the sum of the lengths of its three sides. By solving this equation, we can find the value of the variable and, consequently, the lengths of the two unknown sides. This problem is a classic example of how mathematical ratios and proportions are used in practical geometric calculations, highlighting the interconnectedness of different mathematical concepts. Understanding how to solve such problems is essential not only for academic purposes but also for various real-world applications, such as in architecture, engineering, and design.

Setting Up the Problem

To effectively tackle this problem, we need to translate the given information into a mathematical framework. Let's break down the information we have:

• The perimeter of the triangle is 16 cm.
• One side of the triangle measures 5.5 cm.
• The other two sides are in the ratio 3:4.

Our primary goal is to find the lengths of these two unknown sides. To do this, we'll use the concept of ratios and algebra. Since the two sides are in the ratio 3:4, we can represent their lengths as 3x and 4x, where 'x' is a common multiplier. This representation is crucial because it maintains the proportionality between the sides while allowing us to express them in terms of a single variable. The use of 'x' as a common multiplier is a standard technique in ratio problems, enabling us to convert the ratio into actual lengths. By using this approach, we simplify the problem and set the stage for solving it algebraically. This initial setup is a critical step, as it transforms the geometric problem into an algebraic equation, which we can then solve using standard methods. The beauty of this approach lies in its simplicity and effectiveness in handling problems involving ratios and proportions. Understanding how to set up the problem correctly is often half the battle, and in this case, it paves the way for a straightforward solution.

Forming the Equation

With the sides of the triangle now represented as 5.5 cm, 3x, and 4x, we can proceed to form an equation using the information about the perimeter. The perimeter of a triangle is the sum of the lengths of its three sides. Given that the perimeter is 16 cm, we can write the equation as:

5.  5 + 3x + 4x = 16

This equation is the heart of our problem, as it mathematically connects the known perimeter with the unknown side lengths. The equation states that the sum of the lengths of the three sides (5.5 cm, 3x, and 4x) equals the total perimeter (16 cm). Forming this equation is a critical step because it allows us to use algebraic techniques to solve for 'x'. The equation is linear, which makes it relatively straightforward to solve. Linear equations are a fundamental concept in algebra, and their application in geometry, as seen here, demonstrates the interconnectedness of mathematical fields. The ability to form such equations from given geometric information is a crucial skill in problem-solving. It involves translating a real-world scenario (the triangle and its perimeter) into a mathematical statement that can be manipulated and solved. This skill is not only essential in mathematics but also in various other fields where quantitative problem-solving is required. The equation we have formed is the key to unlocking the solution, and the next step involves simplifying and solving it to find the value of 'x'.

Solving for 'x'

Now that we have the equation 5.5 + 3x + 4x = 16, the next step is to solve for 'x'. This involves simplifying the equation by combining like terms and then isolating 'x' on one side. First, we combine the 'x' terms: 3x + 4x equals 7x. So, the equation becomes:

6.  5 + 7x = 16

Next, we want to isolate the term with 'x'. To do this, we subtract 5.5 from both sides of the equation. This maintains the equality and moves the constant term to the right side:

7x = 16 - 5.5

Performing the subtraction, we get:

8.  x = 10.5

Finally, to solve for 'x', we divide both sides of the equation by 7:

x = 10.5 / 7

x = 1.5

Thus, the value of 'x' is 1.5. This value is crucial because it is the common multiplier we used to represent the lengths of the two unknown sides. Solving for 'x' is a fundamental algebraic skill, and it's a key step in many mathematical problems. The process involves applying basic arithmetic operations while adhering to the rules of algebra, such as maintaining equality by performing the same operation on both sides of the equation. In this case, we used subtraction and division to isolate 'x'. The value of 'x' we have found is not the final answer, but it is a critical intermediate step. It allows us to determine the actual lengths of the two sides of the triangle, which is the ultimate goal of the problem. The next step is to substitute this value of 'x' back into our expressions for the side lengths.

Finding the Side Lengths

Having found the value of x to be 1.5, we can now calculate the lengths of the two unknown sides of the triangle. Recall that we represented these sides as 3x and 4x. To find their lengths, we simply substitute x = 1.5 into these expressions.

For the first side, which is 3x, we have:

9.  * 1.5 = 4.5 cm

So, one side of the triangle is 4.5 cm long.

For the second side, which is 4x, we have:

10.  * 1.5 = 6 cm

Therefore, the other side of the triangle is 6 cm long. These calculations are straightforward but essential for completing the problem. They demonstrate how algebraic solutions can be translated back into geometric terms. The substitution of the value of 'x' is a common technique in algebra, allowing us to find specific values once we have solved for a variable. In this case, it enables us to determine the actual lengths of the sides of the triangle. The two sides we have calculated, 4.5 cm and 6 cm, are the solution to our problem. We have successfully used the given information about the perimeter and the ratio of the sides to find the exact lengths of the unknown sides. This process highlights the power of combining algebraic and geometric concepts to solve problems.

Verifying the Solution

To ensure the accuracy of our solution, it's crucial to verify that the calculated side lengths indeed satisfy the given conditions. We found the two unknown sides to be 4.5 cm and 6 cm. We should check if these lengths, along with the given side of 5.5 cm, add up to the given perimeter of 16 cm, and also if they maintain the given ratio of 3:4.

First, let's check the perimeter. Adding the lengths of the three sides:

11.  5 + 4.5 + 6 = 16 cm

This confirms that the sum of the side lengths matches the given perimeter of 16 cm. Next, we need to verify that the calculated sides (4.5 cm and 6 cm) are indeed in the ratio of 3:4. To do this, we can divide the lengths and see if the ratio holds:

12.  5 / 6 = 0.75

The ratio 3:4 can also be expressed as a decimal by dividing 3 by 4:

13.  / 4 = 0.75

Since both ratios are equal, this confirms that the calculated sides are in the correct proportion. This verification step is vital in problem-solving. It ensures that our solution is not only mathematically correct but also logically consistent with the problem's conditions. The process of verification often involves revisiting the original problem statement and checking if all the conditions are met by the solution. In this case, we checked both the perimeter and the ratio, providing a robust confirmation of our answer. This practice of verifying solutions is a hallmark of careful and thorough problem-solving, and it helps in building confidence in the correctness of the results.

Conclusion

In this article, we successfully solved a geometric problem involving a triangle, its perimeter, and the ratio of its sides. We started by setting up the problem, representing the unknown sides using a variable based on the given ratio. We then formed an equation using the perimeter formula, which we solved to find the value of the variable. This value allowed us to calculate the lengths of the unknown sides. Finally, we verified our solution to ensure its accuracy.

This problem illustrates the power of combining geometric concepts with algebraic techniques. The ability to translate a geometric problem into an algebraic equation is a valuable skill in mathematics and its applications. The problem also highlights the importance of understanding ratios and proportions, as well as the perimeter of a triangle. The steps we followed – setting up the problem, forming an equation, solving for the variable, finding the side lengths, and verifying the solution – are a general approach that can be applied to a wide range of mathematical problems. The key takeaway from this exercise is the interconnectedness of different mathematical concepts and the importance of a systematic approach to problem-solving. By breaking down a complex problem into smaller, manageable steps, we can arrive at a solution with confidence. This approach is not only useful in mathematics but also in various other fields where problem-solving is a crucial skill. The problem we solved is a testament to the beauty and utility of mathematics in understanding and describing the world around us.