These two shapes are similar, and their areas 𝐴 one and 𝐴 two are given. Find the value of 𝑥.
From the diagram, we can see that 𝑥 represents the length in the smaller of the two shapes. We’ve been given the corresponding length in the larger of the two shapes. We’ve also been given both areas: the 40 square inches and 10 square inches.
The key fact that we’re going to use in this question is that these two shapes are similar. This means that there’s a relationship that exists between the ratio of their areas and the ratio of their lengths. The general relationship for similar shapes is this: if the length ratio between the two shapes is 𝑘 to one, meaning that all of the lengths in the larger shape are 𝑘 times the corresponding length in the smaller shape, then the area ratio is 𝑘 squared to one. This means that if we know the length ratio between two similar shapes, we can calculate the area ratio and also vice versa.
In this question, we’ve been given both areas, which means we can calculate the area ratio. The area ratio for these two shapes is 40 to 10. We can simplify this ratio into its simplest form by dividing both sides by 10. So the simplified area ratio is four to one. Now, remember this area ratio can be thought of as 𝑘 squared to one, and we recall that four is equal to two squared. The area ratio can therefore be written as two squared to one.
From this, we can now work out the length ratio. Using the general rule, if the area ratio is two squared to one, then the length ratio is two to one, meaning that all the lengths in the larger shape are twice as long as the corresponding length in the smaller shape. Now, we have all the information we need in order to work out 𝑥. The corresponding side in the larger shape is six inches and remember this is twice as big as 𝑥. Therefore, the value of 𝑥 is six divided by two, which of course is equal to three.
So within this problem, we use the two known areas in order to write down an area ratio. We then use the relationship between the area ratio and the length ratio for similar shapes to find the length ratio and then combine this with the known lengths in the larger shape in order to calculate the corresponding length in the smaller shape, 𝑥.