Finding the Dimensions of a Rectangle Given Its Ratio and Perimeter

When you are given the ratio of the length and width of a rectangle along with its perimeter, you can solve for the exact dimensions using a combination of algebraic and geometric principles. In this article, we will walk through the steps to find the length and width of a rectangle with a given ratio of 5:4 and a perimeter of 72 units.

Understanding the Problem

To start, let's define our variables and the given conditions:

The ratio of the length to the width is 5:4. The perimeter of the rectangle is 72 units. The perimeter (P) of a rectangle is given by the formula: ( P 2 times (length width) ).

Solving the Problem

We will use the formula for the perimeter and the given ratio to find the length and width of the rectangle.

Step 1: Setting Up Equations

Let's say:

Length 5x

Width 4x

Where x is the common factor.

Substituting these into the perimeter formula:

( 72 2 times (5x 4x) )

Simplifying the equation:

( 72 2 times 9x )

( 72 18x )

Step 2: Solving for x

To find x, we divide both sides by 18:

( x frac{72}{18} 4 )

Step 3: Calculating the Dimensions

Now that we have the value of x, we can calculate the exact dimensions:

Length 5x 5 times 4 20 units Width 4x 4 times 4 16 units

Let's verify this by calculating the perimeter again:

( Perimeter 2 times (length width) 2 times (20 16) 2 times 36 72 ) units

This confirms our solution is correct.

Conclusion

Using the given ratio and perimeter, we have successfully determined the dimensions of the rectangle. The length is 20 units and the width is 16 units. This method can be applied to similar problems by following the outlined steps: setting up equations, solving for a variable, and then calculating the dimensions.

Additional Practice and Tips

To further understand and apply these concepts, consider solving similar problems with different ratios and perimeters. Practice is crucial in mastering math and geometry problems. Additionally, always double-check your solutions by plugging the values back into the formulas to ensure accuracy.

Remember, the key to solving such problems is to translate the given information into mathematical equations and solve systematically. With practice, you will become more proficient in handling such geometric problems.

For more information and practice on ratios, perimeters, and solving geometric problems, consider exploring additional resources or practice exercises.