A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increase by 68 cm…

1. Analyze the question

We are dealing with a rectangle where the sides have a specific relationship. We need to find the length of the shorter side by setting up an algebraic equation based on the change in area when the dimensions are increased.

2. Explain the concept

• Area of a rectangle: $Area = \text{length} \times \text{width}$
• Let the shorter side be $x$.
• Since the other side is 6 cm longer, the longer side is $(x + 6)$.
• Original Area = $x(x + 6)$
• When we add 2 cm to each side: • New shorter side = $(x + 2)$ • New longer side = $(x + 6 + 2) = (x + 8)$ • New Area = $(x + 2)(x + 8)$

3. Show the solution

Step 1: Set up the equation. The problem states that the new area is 68 cm² larger than the original area:

Step 2: Expand both sides. Left side: $x^2 + 8x + 2x + 16 = x^2 + 10x + 16$ Right side: $x^2 + 6x + 68$

Step 3: Solve for $x$.

Subtract $x^2$ from both sides:

Subtract $6x$ from both sides:

Subtract 16 from both sides:

Divide by 4:

The shorter side is 13 cm.

4. Identify the answer

Comparing our result to the options provided: A. 15 cm B. 19 cm C. 13 cm D. 21 cm

The correct option is C.

5. Tip

When dealing with "area increase" problems, always write out the "Before" and "After" expressions for the area separately before equating them. This prevents errors in expansion. Also, remember that adding 2 cm to each side means adding 2 to both the length and the width!