A rectangle has width and length . The rectangle is cut along the horizontal and vertical dotted lines to produce four smaller rectangles. The sum of the perimeters of these four rectangles is 24. What is the value of ?
Problem 322
Official solution
The sum of the lengths of the horizontal line segments is , because the tops of the four small rectangles contribute a total of to their combined perimeter and the bottoms contribute a total of . Similarly, the sum of the lengths of the vertical line segments is . In other words, the sum of the perimeters of the four rectangles is . Since the sum of the perimeters also equals 24, then and so .