If the two following polygons are similar, find the value of 𝑥.
We can solve for 𝑥 using proportions. And we can set this up two different ways. And we are able to do this because we know the polygons are similar. One way to set up their proportion is to compare the larger polygon to the smaller polygon by putting the larger polygon’s dimensions on the numerators and the smaller polygon’s dimensions on the denominators.
Here we’ve written the larger polygon’s dimensions on the numerators. Now we have to be careful. When we write the smaller polygon’s dimensions on the denominators, since we began with the 75 on the first fraction, we need to begin with the 𝑥 on the first fraction for the smaller polygon, just like this.
And the way that we will solve is by cross-multiplying. So let’s take 𝑥 times 85 to get 85𝑥 and 75 times 34 to get 2550. So to solve for 𝑥, we divide both sides by 85. And we get that 𝑥 is equal to 30.
Now let’s set this up a different way. The other way to set this up is the corresponding sides. And we’ve go ahead and we’ve put in the top lengths, which are the 75 and the 𝑥, and then the side lengths, the 85 and 34.
Again, the numbers that correspond on the top are 75 and 𝑥. Now since we started with 75, we need to start with 85 for our denominator and 34 for the second denominator, because the 75 came from the larger polygon. So the 85 needs to come from the larger polygon, the one for that same fraction.
And same with the second fraction, they both come from the smaller polygon. So cross-multiplying, we have 75 times 34, which is 2550. And we set it equal to 85 times 𝑥, so 85𝑥. So now we divide both sides by 85. And we get 30 is equal to 𝑥. So once again, the value of 𝑥 would be 30, making that side length 30 centimeters.