Square A is an enlargement of square B by a scale factor of two-thirds. If the perimeter of square A equals 56 centimeters, what is the area of square B? Give your answer to the nearest hundredth.
Let’s begin by drawing a sketch to help us visualize the problem. We’re told that square A is an enlargement of square B by a scale factor of two-thirds, which does in fact mean that square A will be smaller than square B. If we want to draw on the arrow representing the enlargement, then because square A is an enlargement of square B by a scale factor of two-thirds, then the arrow would go in this direction.
Next, let’s see what information we’re given about the measurements of these squares. Well, we’re given that the perimeter of square A is 56 centimeters. And we need to work out the area of square B. Perimeter and area won’t be directly related, but let’s see if we can work out the length of the sides of either square A or square B.
As we know that these two shapes are squares, we can use the perimeter of square A to work out its side length. We can remember that a square of side length 𝑥 will have a perimeter of four 𝑥. And so, if we take the length of square A to be 𝑥 centimeters, then we can say that four 𝑥 must be equal to 56. Dividing both sides by four then, 56 divided by four would give us a value of 𝑥 as 14, which means we have now worked out that the side length of square A is 14 centimeters.
We can then use the enlargement scale factor to help us work out the side length of square B. We know that the enlargement scale factor from B to A is two-thirds. To reverse this, we will divide by two-thirds, which is equivalent to multiplying by the reciprocal three over two. Therefore, to work out the side length of square B, we will take the side length of square A, which is 14, and multiply it by three over two. This simplifies to 42 over two. And working that out gives us a value of 21 centimeters. And so, the corresponding side length in the similar shape of square B would be 21 centimeters.
Of course, because it’s a square, the corresponding side could be any side in square B. Now that we have the side length of square B, it’s relatively simple to work out its area using the fact that the area of a square of side length 𝐿 is 𝐿 squared. And so, we can write that the area of square B is 21 squared. We can then work this out as 441. 441 is an integer, but we were asked to give our answer to the nearest hundredth. And we can indicate this by adding two decimal digits. We can then complete the answer with the units. And so, the area of square B is 441.00 square centimeters.