𝐴𝐵𝐶𝐷 is similar to 𝐸𝐹𝐺𝐻, where 𝐴𝑁 equals seven centimeters and 𝐸𝑀 equals 2.8 centimeters. If the area of 𝐴𝐵𝐶𝐷 is 1,848 square centimeters, what is the area of 𝐸𝐹𝐺𝐻?
We are given the information that the two shapes in the figure are similar. And we remember that similar shapes have the same number of sides and their corresponding angles are congruent and their corresponding sides are in the same proportion. If we label the information that 𝐴𝑁 equals seven centimeters and 𝐸𝑀 equals 2.8 centimeters onto the figure, then we can observe that these are two corresponding sides. This means that we can write the ratio of the lengths between 𝐴𝐵𝐶𝐷 and 𝐸𝐹𝐺𝐻 as seven to 2.8.
There is then a way in which we can relate the ratio of the lengths of two similar shapes to the ratio of their areas. If the length ratio of two similar shapes is 𝑎 to 𝑏, then the ratio of their areas is 𝑎 squared to 𝑏 squared. And so we can write the area ratio of 𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻 as seven squared to 2.8 squared. We can then evaluate the squares to give us an area ratio of 49 to 7.84.
Now, this doesn’t mean that the areas will be exactly 49 square centimeters or 7.84 square centimeters. But rather their areas will be equivalent to those given in the area ratio. We are in fact told that the area of 𝐴𝐵𝐶𝐷 is 1,848 square centimeters. And so if we define the unknown area of 𝐸𝐹𝐺𝐻 as 𝑥, that means that we need to find the value of 𝑥 such that the ratio 49 to 7.84 and 1,848 to 𝑥 are equivalent. One way to do this is to consider what value we must multiply 49 by to get 1,848. We would then multiply 7.84 by that same value, which would give us 𝑥.
When we use our calculators to work out 1,848 divided by 49, we get a long recurring decimal of 37.714 and so on. Alternatively, as a fraction, we can give this as 264 over seven. Using this fractional value, we can multiply 7.84 by 264 over seven. If you are using a decimal value, keep that value in the calculator and multiply it by 7.84. And therefore, we get the answer of 295.68.
Another way to find this value of 𝑥 would be to apply our fractional reasoning. We know that 49 over 1,848 is in the same proportion as 7.84 over 𝑥. We would then cross multiply and divide both sides by 49. Performing this calculation on our calculators would give us the same value of 295.68. So either of these methods for working out the equivalent ratio value would give us the answer that the area of 𝐸𝐹𝐺𝐻 is 295.68 square centimeters.