Problem 27 - Elementary Algebra

Question

Suppose that the perimeter of a square equals the perimeter of a rectangle. The width of the rectangle is 9 inches less than twice the side of the square, and the length of the rectangle is 3 inches less than twice the side of the square. Find the dimensions of the square and the rectangle.

Step-by-Step Solution

Answer

The square has a side of 6 inches; the rectangle is 3 inches wide and 9 inches long.

Step 1: Formula for Perimeter of Square

The formula for the perimeter of a square is given by the expression: \[ P_{square} = 4s \] where \( s \) is the length of a side of the square.

Step 2: Formula for Perimeter of Rectangle

The formula for the perimeter of a rectangle is given by: \[ P_{rectangle} = 2(l + w) \] where \( l \) is the length and \( w \) is the width of the rectangle.

Step 3: Set Perimeters Equal

Since the perimeters of the square and the rectangle are equal: \[ 4s = 2(l + w) \]

Step 4: Express Length and Width

According to the problem, the width of the rectangle is \( w = 2s - 9 \), and the length is \( l = 2s - 3 \). Substitute these into the perimeter equation.

Step 5: Substitute Values into Perimeter Equation

Substitute \( l = 2s - 3 \) and \( w = 2s - 9 \) into the perimeter equation: \[ 4s = 2((2s - 3) + (2s - 9)) \]

Step 6: Simplify the Equation

Simplify the equation: \[ 4s = 2(4s - 12) \] which simplifies to \[ 4s = 8s - 24 \]

Step 7: Solve for 's'

Rearrange the equation to solve for \( s \):\[ 24 = 8s - 4s \] or \[ 24 = 4s \] Divide both sides by 4 to get:\[ s = 6 \]

Step 8: Find Dimensions of Rectangle

Now that we found \( s = 6 \), use this to find the dimensions of the rectangle:- Width \( w = 2(6) - 9 = 12 - 9 = 3 \)- Length \( l = 2(6) - 3 = 12 - 3 = 9 \)

Key Concepts

Perimeter of square

To find the perimeter of a square, you use the formula: \( P_{square} = 4s \), where \( s \) is the length of one side of the square. This formula works because all four sides of a square have the same length, so multiplying the length of one side by 4 gives you the total perimeter. This concept is straightforward, but it's important to understand because many geometric problems use the perimeter of a square as a fundamental component. Remember that knowing just one side's length allows you to determine the entire perimeter of the square.

Perimeter of rectangle

The perimeter of a rectangle can be found using the formula: \( P_{rectangle} = 2(l + w) \), where \( l \) represents the length and \( w \) represents the width of the rectangle. This formula sums the length and width and then multiplies by 2, reflecting the fact that a rectangle has two sets of parallel sides.Understanding this formula is essential for solving problems where you need to compare or adjust the dimensions of a rectangle.

Algebraic equations

Algebraic equations are used to solve problems by finding unknown values. In this exercise, the equation \( 4s = 2(l + w) \) was fundamental.It represented the scenario where a square and a rectangle have identical perimeters, which allowed us to use algebra to express the dimensions of the rectangle in terms of the square's side length. By solving these equations step by step, you can find unknown dimensions by logically arranging and simplifying the expressions.

Geometry concepts

Geometry concepts involve understanding shapes, sizes, and the properties of space. In this problem, we dealt with squares and rectangles—two basic geometric figures. The definitions and properties of these shapes are crucial:

• A square has four equal sides, making calculations straightforward.
• A rectangle, however, has two sets of equal sides, requiring different formulaic approaches.

These foundational concepts enable us to set up equations correctly, especially when working with measurements and dimensions. Mastering basic geometry builds the groundwork needed to tackle more complex problems efficiently.