Solving Rectangle Dimensions Using Equations and Perimeter Formula

The perimeter of a rectangle is a key concept in geometry that helps us understand the sum of the lengths of its sides. By using algebraic equations, we can solve for unknown dimensions of a rectangle given its perimeter and a relationship between its length and width. In this article, we will solve three examples to illustrate this concept.

Example 1: Perimeter of 52 Inches

In the first example, the length of a rectangle is 6 inches less than 3 times the width, and the perimeter is 52 inches. We need to find the length and width.

Step-by-Step Solution:

Define the Variables: Let w be the width of the rectangle. Express Length in Terms of Width: The length l can be expressed as l 3w - 6. Use the Perimeter Formula: Perimeter 2l 2w. Set Up the Equation: 2(3w - 6) 2w 52 Simplify the Equation: 6w - 12 2w 52 Combine Like Terms: 8w - 12 52 Isolate the Variable: 8w 64 Solve for w: w 8 inches Find the Length: l 3w - 6 3(8) - 6 24 - 6 18 inches

Therefore, the dimensions of the rectangle are: Width: 8 inches Length: 18 inches

Example 2: Perimeter of 62 Yards

For the second example, the length is five yards less than three times the width, and the perimeter is 62 yards. We need to find the length and width.

Step-by-Step Solution:

Define the Variables: Let W be the width of the rectangle. Express Length in Terms of Width: The length L can be expressed as L 3W - 5. Use the Perimeter Formula: Perimeter 2L 2W. Set Up the Equation: 2(3W - 5) 2W 62 Simplify the Equation: 6W - 10 2W 62 Combine Like Terms: 8W - 10 62 Isolate the Variable: 8W 72 Solve for W: W 9 yards Find the Length: L 3W - 5 3(9) - 5 27 - 5 22 yards

Therefore, the dimensions of the rectangle are: Width: 9 yards Length: 22 yards

Example 3: Perimeter of 62 Yards

In the third example, the length is five yards less than three times the width, and the perimeter is 62 yards. We need to find the length and width.

Step-by-Step Solution:

Define the Variables: Let W be the width of the rectangle. Express Length in Terms of Width: The length L can be expressed as L 3W - 5. Use the Perimeter Formula: Perimeter 2L 2W. Set Up the Equation: 2(3W - 5) 2W 62 Simplify the Equation: 6W - 10 2W 62 Combine Like Terms: 8W - 10 62 Isolate the Variable: 8W 72 Solve for W: W 9 yards Find the Length: L 3W - 5 3(9) - 5 27 - 5 22 yards

Therefore, the dimensions of the rectangle are: Width: 9 yards Length: 22 yards

Conclusion

Understanding how to solve for missing dimensions of a rectangle using the perimeter formula and algebraic equations is a fundamental skill in geometry. By following these steps and examples, you can easily find the missing measurements of rectangles provided certain conditions.