The length of a rectangle is 7 more than its width. If the perimeter of the rectangle is the same as the perimeter of a square of side 8.5, what is the length of a diagonal of the rectangle?

To understand why the answer is 13, let's break down the problem step by step:

    
• Step 1: Find the perimeter of the square.         

  The perimeter of a square is calculated by the formula P = 4 × side. For a square with a side of 8.5, the perimeter is:

          

  P = 4 × 8.5 = 34

      
    
• Step 2: Set up the equations for the rectangle.         

  Let the width be represented as w. According to the question, the length is l = w + 7.

          

  The formula for the perimeter of a rectangle is P = 2 × (length + width). Given the perimeter is 34, we can write:

          

  34 = 2 × (w + (w + 7))

      
    
• Step 3: Solve for width.         

  Expanding the equation: 34 = 2 × (2w + 7)

          

  This simplifies to 34 = 4w + 14, leading to 20 = 4w, which gives w = 5.

      
    
• Step 4: Find the length of the rectangle.         

  Using the width found, the length will be:

          

  l = 5 + 7 = 12

      
    
• Step 5: Calculate the diagonal of the rectangle.         

  The diagonal can be found using the Pythagorean theorem: d = √(length² + width²).

          

  Substituting the values gives:

          

  d = √(12² + 5²) = √(144 + 25) = √169 = 13

      

Thus, the correct answer is 13, which represents the length of the diagonal of the rectangle.

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