Problem number 17. 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? So first we will remember this fact from geometry, that the perimeter of a rectangle is 2L plus 2W, where L and W are the length and the width. Let's imagine gluing back our four small rectangles into the large rectangle again. And let's look at the perimeters of how some of these rectangles relate to each other. For example, if we look at the rectangles with perimeters 16 and 18, well the rectangle with perimeter 16 also has perimeter 2A plus 2D, because we notice that its dimensions are A and D, as shown on the diagram. Similarly, the rectangle with perimeter 18 has dimensions B and C, so its perimeter is 2B plus 2C. If we also look at the rectangles with perimeters question mark and 24, the rectangle with perimeter question mark has perimeter 2A plus 2C, and the rectangle with perimeter 24 has dimensions B by D, that means that its perimeter is 2B plus 2D. And we notice that both of these values are equal to 2A plus 2B plus 2C plus 2D. This is because addition is commutative, we can rearrange the terms in each addition however we want, and notice that each of the letters A, B, C, D appears exactly once in both of those sentences above. That means that the sum of the perimeters 16 and 18 is equal to the sum of the perimeters question mark and 24, since they're both equal to the same thing. And solving for question mark, we get that the question mark equals 16 plus 18 minus 24, which is equal to 10. That means that the answer is B, 10.

rectangle

perimeter

geometry

calculation

mathematics