Solving Rectangle Geometry Problems

We are given a rectangle where the longer side is 2 cm greater than the shorter side. The area of the rectangle is 80 cm². We need to find the perimeter of this rectangle.

Setting up the Rectangle Dimensions

Let's denote the shorter side of the rectangle as \(w\) cm.

According to the problem, the longer side (\(l\)) is 2 cm greater than the shorter side. So, we can express the longer side as \(l = w + 2\) cm.

Using the Area to Find Side Lengths

The area of a rectangle is calculated by multiplying its length and width (Area = \(l \times w\)).

We are given that the area is 80 cm². Substituting the expressions for \(l\) and \(w\), we get the equation:

\(w \times (w + 2) = 80\)

Now, let's solve this equation for \(w\):

\(w^2 + 2w = 80\)

To solve this quadratic equation, we move all terms to one side:

\(w^2 + 2w - 80 = 0\)

We can solve this quadratic equation by factoring. We need to find two numbers that multiply to -80 and add up to +2. These numbers are 10 and -8.

So, we can factor the equation as:

\((w + 10)(w - 8) = 0\)

This equation gives us two possible values for \(w\):

• \(w + 10 = 0 \Rightarrow w = -10\)
• \(w - 8 = 0 \Rightarrow w = 8\)

Since \(w\) represents the length of a side of the rectangle, it must be a positive value. Therefore, we take the positive solution:

\(w = 8\) cm

Calculating Rectangle Side Lengths

The shorter side \(w = 8\) cm.

The longer side \(l = w + 2 = 8 + 2 = 10\) cm.

So, the dimensions of the rectangle are 8 cm and 10 cm.

Let's quickly check the area: \(8 \times 10 = 80\) cm², which matches the given information.

Calculating the Rectangle Perimeter

The perimeter of a rectangle is calculated using the formula: Perimeter = \(2 \times (\text{length} + \text{width})\).

Using the side lengths we found (\(l = 10\) cm and \(w = 8\) cm):

Perimeter = \(2 \times (10 + 8)\)

Perimeter = \(2 \times (18)\)

Perimeter = \(36\) cm

The perimeter of the rectangle is 36 cm.

Revision Table: Key Formulas

Additional Information: Properties of Rectangles

A rectangle is a quadrilateral with four right angles. Key properties include:

• Opposite sides are equal in length (\(l_1 = l_2\), \(w_1 = w_2\)).
• Opposite sides are parallel.
• Diagonals are equal in length and bisect each other.
• The sum of interior angles is 360 degrees. Each angle is 90 degrees.

Understanding these properties helps in solving various geometry problems involving rectangles.