# Lesson 3

Types of Transformations

• Let’s analyze transformations that produce congruent and similar figures.

### 3.1: Why is it a Dilation?

Point $$B$$ was transformed using the coordinate rule $$(x,y) \rightarrow (3x,3y)$$.

1. Add these auxiliary points and lines to create 2 right triangles: Label the origin $$P$$. Plot points $$M=(2,0)$$ and $$N=(6,0)$$. Draw segments $$PB',MB,$$ and $$NB’$$.
2. How do triangles $$PMB$$ and $$PNB’$$ compare? How do you know?
3. What must be true about the ratio $$PB:PB'$$?

### 3.2: Congruent, Similar, Neither?

Match each image to its rule. Then, for each rule, decide whether it takes the original figure to a congruent figure, a similar figure, or neither. Explain or show your reasoning.

1. $$(x,y) \rightarrow \left(\frac{x}{2}, \frac{y}{2}\right)$$
2. $$(x,y) \rightarrow (y, \text-x)$$
3. $$(x,y) \rightarrow (\text-2x, y)$$
4. $$(x,y) \rightarrow (x-4, y-3)$$

Here is triangle $$A$$.

1. Reflect triangle $$A$$ across the line $$x=2$$.
2. Write a single rule that reflects triangle $$A$$ across the line $$x=2$$.

### 3.3: You Write the Rules

1. Write a rule that will transform triangle $$ABC$$ to triangle $$A’B’C’$$.
2. Are $$ABC$$ and $$A’B’C’$$ congruent? Similar? Neither? Explain how you know.
3. Write a rule that will transform triangle $$DEF$$ to triangle $$D’E’F’$$.
4. Are $$DEF$$ and $$D’E’F’$$ congruent? Similar? Neither? Explain how you know.

### Summary

Triangle $$ABC$$ has been transformed in two different ways:

• $$(x,y) \rightarrow (\text-y,x)$$, resulting in triangle $$DEF$$
• $$(x,y) \rightarrow (x,3y)$$, resulting in triangle $$XYC$$
Let’s analyze the effects of the first transformation. If we calculate the lengths of all the sides, we find that segments $$AB$$ and $$DE$$ each measure $$\sqrt5$$ units, $$BC$$ and $$EF$$ each measure 5 units, and $$AC$$ and $$DF$$ each measure $$\sqrt{20}$$ units. The triangles are congruent by the Side-Side-Side Triangle Congruence Theorem. That is, this transformation leaves the lengths and angles in the triangle the same—it is a rigid transformation.
Not all transformations keep lengths or angles the same. Compare triangles $$ABC$$ and $$XYC$$. Angle $$X$$ is larger than angle $$A$$. All of the side lengths of $$XYC$$ are larger than their corresponding sides. The transformation $$(x,y) \rightarrow (x,3y)$$ stretches the points on the triangle 3 times farther away from the $$x$$-axis. This is not a rigid transformation. It is also not a dilation since the corresponding angles are not congruent.