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 \((x,y) \rightarrow (3x,3y)\).

B=2 comma 3. B prime= 6 comma 9.
  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.

A

Two rectangles on coordinate plane.
BTriangles F and F prime on coordinate plane.

C

Triangles F and F prime on coordinate plane.

D

Graph of triangles F and F prime. 
  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 \(A\).

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

4 triangles on coordinate plane.
  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 \(XYC\)
3 triangles on coordinate plane.

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.