Similarity & Proofs
Similarity is an incredibly helpful mathematical tool. Just ask the folks that make maps or any skilled painter. The properties of dilated figures make it possible for us to express the world as we see it: a scale model of reality or a glimpse outward to the horizon.
Just as “almost” is not the same as “totally”, similarity is not the same as congruence. Here, we look at the varying levels of “same” and what properties are shared by these similar figures.
We’ll even look at a few situations that aren’t very “same” at all! And, as always, we’ll PROVE IT!
Dilations & Properties of Similarity
So, we’ve looked at the isometric transformations–the ones that result in exact, congruent copies of the original. Not all transformations are isometric, but they are all predictable.
Knowing that a dilation “multiplies the coordinates” or “changes the size of a figure” is important, but it does little to predict the properties of a figure that’s been dilated.
In this topic, we go beyond the ‘textbook’ rules to explore in detail what happens when figures are dilated.
Here, we return to the idea of 2-column proofs. This time, in addition to looking for congruent angles, we will be looking to prove the other properties of dilation — that the corresponding sides are proportional.
Visiting a few math theorems along the way, we’ll add a whole new chapter to our Proof Writing credentials.
Solving with Similarity
Setting up ratios and proportions are not new. But now, we have a more refined set of tools. We can apply these concepts to real-world problems.
May you never be stranded in a situation where you need to calculate the height of a flagpole, based on the size of its shadow. . . but if you find yourself in the situation, you’ll know how to do it!