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Geometry Session 8: Similarity
 
Session 8 Part A Part B Part C Homework
 
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Session 8, Part A:
Scale Drawings

In This Part: Finding a "Good Copy" | Doubling the Coordinates | Distances and Angles

Why does the coordinate trick work to create similar figures?

Rather than looking at the head of a cat, let's look at a triangle to explain why multiplying the coordinates by 2 creates a similar figure. We need to show two things:

1. 

The distance between any two points doubles.

2. 

The angles all stay the same.

Since we know that we can take any polygon and split it up into triangles, this will be enough to show that the scaling trick works for any polygon. If the triangles in the new polygon are all similar to the triangles in the original, then the two polygons themselves must be similar.

Here are two arbitrary points, A = (a1,a2) and B = (b1,b2), on a coordinate grid:

Problem A6

Solution  

What is the distance between the two points? (Look back to Session 6 if you don't remember how to find the distance. It might also help to try some specific examples.)

 

Problem A7

Solution  

If you apply the rule (x,y) (2x,2y) to both points to create A' and B', what are the new coordinates? What is the distance between A' and B'? Are you sure that it's double the distance from A to B?

 
 

The reason that the angles don't change is a little trickier. Here's the idea: There is some triangle DEF, similar to triangle ABC, but with sides twice as long. (Imagine putting ABC in a copy machine and enlarging it by 100%.) The corresponding angles all have the same measures: mA = mD, mB = mE, and mC = mF.

Because triangles are rigid (SSS congruence), any triangles with the same sides as DEF will also have the same angles as DEF.

In particular, the triangle made by doubling the coordinates of A, B, and C has sides twice as long as triangle ABC. That is, the sides are the same as DEF. So the angles are also the same as DEF and as ABC.

That means the sides are in proportion to the sides of ABC (they're twice as long) and the angles have the same measures. So the two triangles are similar.

This argument is what we call general in principle. We explained why doubling the coordinates produces similar triangles. But the same argument would work if you multiplied the coordinates by any number.


Take it Further

Problem A8

 Solution

How could you modify the definition of similar polygons to test whether figures like the two below are similar?

 

Next > Part B: Similar Triangles

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