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Similarity and Area Ratios - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

If two triangles are similar, then their corresponding sides are proportional. Since sides are a length and lengths are one dimensional, the side ratio will not predict the ratio of the areas. To find the area ratios, raise the side length ratio to the second power. This applies because area is a square or two-dimensional property.

We can use this idea of similarity and apply it to area. Let's say we had 2 polygons that are similar. So here we have abcd is similar to efgh. And if I picked one of their corresponding sides that is ab and ef, I know that that ratio is going to be constant for all of the corresponding sides. That ratio will be a:b, or whatever those numbers are for their actual lengths.
Now what is the ratio of their areas going to be? Well, to examine this, let's move over here and talk about dimensionality. Something that is one dimensional is just a length. So think of a distance. Distances we would say have units of centimetres, inches, miles, but it's a one dimensional attribute. Here in these two, the lengths are a distance so the ratio of their one dimensional attribute is a:b. So let's go back and write that ratio for one dimensionality.
Now two dimensionality is talking about area, so we would talk about area as something to the second power. You think about the area of a square we're going to have side times side, s times s is s squared. So to go from one dimensional to two dimensional, we take our exponent and we're going to square it. So in two dimensions the ratio of their areas will be that ratio in one dimension squared. Or we could say this is a squared to b squared. So to go from one dimension to two dimensions, you're going to square whatever your ratio is. To go in the opposite direction you're going to have to do the opposite of squaring something which is the square root.
Let's look at a brief example of how we can apply this. Here it says if two similar triangles have corresponding medians in a ratio of 3:5, what is the ratio of their areas. The first thing that will trip up students about this statement is medians. Well you have to remember that if you have corresponding medians in similar triangles, that they're going to be proportional. So just because we're talking about medians, doesn't change this ratio of 3:5. Median is a one dimensional attribute of this triangle, it's a distance. So this ratio of 3:5, I can write under the category of one dimensional. So to go from one dimensions to two dimensions cause it's asking about area. So we want something's that two dimensional. We know and actually I don't know what that number is. It's supposed to be 3:5. To go from one dimensions to two dimensions we need to square our ratio. So we're going to take three fifths and square it. So 3 squared is 9, 5 squared is 25.
So notice that the one dimensional ratio is 3:5 and the ratio of their areas will be 9:25. This is not saying that the ratio is 9, or excuse me that the area is 9 and the area is 25, it's just saying that when you write a ratio of there is it will be 9:25.

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