Let's go through this worksheet step-by-step. Here's how we'll solve it:

1. Rectangle DEFG:

• Vertices: $D(-2, 7), E(2, 3), F(0, 1), G(4, 5)$.

• a. Translation Rule: $(x, y) \to (x + 6, y - 8)$. Apply the rule to each vertex:

  • $D(-2, 7) \to (4, -1)$.
  • $E(2, 3) \to (8, -5)$.
  • $F(0, 1) \to (6, -7)$.
  • $G(4, 5) \to (10, -3)$.

• b. Reflection Across the $y$-Axis: Reflect the translated points $(x, y) \to (-x, y)$:

  • $D'(4, -1) \to D''(-4, -1)$.
  • $E'(8, -5) \to E''(-8, -5)$.
  • $F'(6, -7) \to F''(-6, -7)$.
  • $G'(10, -3) \to G''(-10, -3)$.

2. Triangle LMN:

• Vertices: $L(6, 6), M(8, 8), N(8, 3)$.

• a. Reflection in the Line $x = 5$: Reflect each point over $x = 5$. For a point $(x, y)$, the reflected point is $(2 \times 5 - x, y)$:

  • $L(6, 6) \to L'(4, 6)$.
  • $M(8, 8) \to M'(2, 8)$.
  • $N(8, 3) \to N'(2, 3)$.

• b. $270^\circ$ Counterclockwise Rotation Around the Origin: For a $270^\circ$ counterclockwise rotation, use the rule $(x, y) \to (y, -x)$:

  • $L(6, 6) \to L''(6, -6)$.
  • $M(8, 8) \to M''(8, -8)$.
  • $N(8, 3) \to N''(3, -8)$.

3. Quadrilateral ABCD:

• Vertices: $A(0, 6), B(-3, -6), C(-9, -6), D(-12, -3)$.

• a. Dilation with a Scale Factor of $\frac{1}{3}$, Centered at the Origin: For a dilation, multiply each coordinate by $\frac{1}{3}$:

  • $A(0, 6) \to A'(0, 2)$.
  • $B(-3, -6) \to B'(-1, -2)$.
  • $C(-9, -6) \to C'(-3, -2)$.
  • $D(-12, -3) \to D'(-4, -1)$.

• b. Translation Rule: $(x, y) \to (x - 5, y - 1)$. Apply the rule to the dilated points:

  • $A'(0, 2) \to A''(-5, 1)$.
  • $B'(-1, -2) \to B''(-6, -3)$.
  • $C'(-3, -2) \to C''(-8, -3)$.
  • $D'(-4, -1) \to D''(-9, -2)$.

Let me know if you need further clarification or have additional questions!

5 Related Questions:

1. What happens to a point when reflected across the $y$-axis?
2. How is dilation different from translation in transformations?
3. What is the general rule for rotating a point $180^\circ$ around the origin?
4. Can you determine if two transformations commute? If not, why?
5. How would translating and reflecting a figure impact its shape and size?

Tip:

Always label and plot points step-by-step to avoid confusion in transformations.