13. Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16,18 and 24 , as shown in the diagram.
What is the perimeter of the fourth small rectangle?
A 8
B 10
C 12
D 14
E 16
Solution
B

Consider the two rectangles with perimeter 18 and 16. It can be seen in the diagram that the sum of the lengths of horizontal sides of these rectangles is equal to the sum of the lengths of the horizontal sides of the large rectangle. Similarly the sum of the lengths of the vertical sides of these two rectangles is equal to the lengths of the vertical sides of the large rectangle. Hence the perimeter of the large rectangle is equal to the sum of the perimeters of these two rectangles. A similar argument can be applied to the rectangle with perimeter 24 and the rectangle with the unknown perimeter. Therefore $18+16=24+P$, where $P$ is the unknown perimeter. Hence the perimeter of the fourth small rectangle is 10 .