The Sides of a Triangle Given Its Ratio and Perimeter

Often in geometry, we are given the ratio of the sides of a triangle along with its perimeter, and we need to find the actual lengths of the sides. This problem is a great example of how to apply fundamental concepts of ratios and perimeters to find the solution.

Simplifying the Ratio

When dealing with ratios such as 1/2 : 1/3 : 1/4, it may seem complicated at first glance. However, simplifying these ratios can make the problem much easier to solve. For instance, let's consider the ratio 1/2 : 1/3 : 1/4 and simplify it to 6:4:3. This simplification is valid because when we find the least common multiple (LCM) of the denominators (2, 3, and 4), which is 12, we can multiply each term in the ratio by 12.

Simplification Using LCM

By multiplying each part of the ratio with 12, we get:

1/2 × 12 : 1/3 × 12 : 1/4 × 12 6 : 4 : 3

Using the Given Perimeter

Now, we know that the perimeter of the triangle is 78 cm, and the sides are in the ratio 6:4:3. This means the total of the coefficients (6 4 3) of the ratio represents the entire perimeter. Therefore, 6x 4x 3x 78, where x is a common multiplier.

The sum of the coefficients is:

6 4 3 13

So, the value of x can be found by setting up the equation:

13x 78

Solving for x, we get:

x 78 / 13 6

Calculating the Side Lengths

Now that we have the value of x, we can find the lengths of each side:

The first side is 6x 6 × 6 36 cm The second side is 4x 4 × 6 24 cm The third side is 3x 3 × 6 18 cm

Verification

To verify our solution, we can add the three sides together and check that their sum is indeed the perimeter given in the problem. Here's the verification:

36 cm 24 cm 18 cm 78 cm

This confirms that our solution is correct.

Summary

The sides of the triangle are 36 cm, 24 cm, and 18 cm, respectively. Understanding how to convert complex ratios into simpler forms and using the given perimeter to find the side lengths is a valuable skill in solving geometric problems. This problem serves as a good exercise in applying mathematical operations to real-world problems.

Additional Tips

When dealing with such problems, remember to:

Convert ratios to their simplest form Identify the common multiplier (x) using the perimeter Calculate each side's length by multiplying the simplified parts of the ratio by the common multiplier Verify your solution by adding the calculated side lengths to ensure they match the given perimeter.