Problem 262 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. Find the length of a rectangle with perimeter 20.2 and width 7.8 .

Step-by-Step Solution

Answer

The length of the rectangle is 2.3 units.

Step 1: Understand Rectangle Properties

A rectangle has two pairs of equal sides. The perimeter of a rectangle is calculated by the formula: Perimeter = 2(Length + Width).

Step 2: Identify Given Information

The problem provides the perimeter and width of the rectangle. Perimeter = 20.2 units and Width = 7.8 units.

Step 3: Write the Perimeter Formula

Use the perimeter formula for a rectangle: Perimeter = 2(Length + Width). Substitute the given values into the formula.

Step 4: Substitute Values into the Formula

Substituting the given values: 20.2 = 2(Length + 7.8).

Step 5: Solve for Length

Divide both sides of the equation by 2 to isolate the Length: 10.1 = Length + 7.8. Then subtract 7.8 from both sides to solve for Length: Length = 10.1 - 7.8 = 2.3 units.

Key Concepts

Rectangle Properties

A rectangle is a special type of quadrilateral with four right angles. It has two pairs of opposite sides that are equal in length. This is crucial because it simplifies the calculations related to the rectangle. For example, if you know the length of one pair of sides, you automatically know the length of the other pair. This symmetry helps in using the perimeter and area formulas effectively.
Furthermore, rectangles have these important properties:

• All angles are 90 degrees.
• Opposite sides are equal in length.
• The diagonals are equal in length and bisect each other.

Perimeter Formula

The perimeter of a shape is the total distance around its edges. For rectangles, you add up all four sides. Because a rectangle has two pairs of equal sides, there is a simpler formula for calculating its perimeter.
The formula to calculate the perimeter (\text{P}) of a rectangle is: \[\text{Perimeter} = 2(\text{Length} + \text{Width})\]
Here’s a brief explanation of how the formula works:

• The 2 on the left side of the equation accounts for both pairs of equal sides.
• The Length and Width inside the parentheses represent the two different side lengths.

With this formula, you can find the perimeter if you know the length and width, or you can find an unknown side length if you have the perimeter. This is especially useful in exercises like the one given where you need to solve for one of the sides.

Solving Equations

Solving equations is a fundamental skill in algebra and geometry. In this exercise, we use the given values and the known perimeter formula to solve for an unknown length. Here’s a quick guide to solving such equations step-by-step:

• **Step 1:** Substitute the known values into the perimeter formula. For example, if the perimeter is 20.2 units and the width is 7.8 units, the equation would look like this:
  \[20.2 = 2(\text{Length} + 7.8)\]
• **Step 2:** Simplify the equation. First, divide both sides by 2 to isolate the Length and Width term:
  \[10.1 = \text{Length} + 7.8\]
• **Step 3:** Solve for the unknown variable (Length). Subtract the width from both sides:
  \[\text{Length} = 10.1 - 7.8\]
• **Step 4:** Calculate the result. Performing the subtraction gives you:
  \[\text{Length} = 2.3\]

Following these steps makes it straightforward to find the length of the rectangle using the given perimeter and width.