Lesson 3

Types of Transformations

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

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

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 $NB’$ .
How do triangles $PMB$ and $PNB’$ compare? How do you know?
What must be true about the ratio $PB:PB'$ ?

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.

Triangles F and F prime on coordinate plane.

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

Are you ready for more?

Here is triangle $A$ .

Triangle A with endpoints 2 comma 2, 8 comma 4, and 6 comma 8.

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

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

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.

Lesson 3

3.1: Why is it a Dilation?

3.2: Congruent, Similar, Neither?

3.3: You Write the Rules

Summary