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Questions and Solutions

6. Name the line of reflection from D to E.

To find the line of reflection:

• Look for the line equidistant from points D and E.
• Based on the grid, the line of reflection appears to be $x = 1.5$.

7. What is the rule for the translation from F to D?

Translation rule:

• From F to D, calculate the movement along the $x$- and $y$-axes:
  • $F(1, 5) \to D(-2, 4)$
  • $x: 1 \to -2$ (subtract 3), $y: 5 \to 4$ (subtract 1).
• The translation rule is: $(x, y) \to (x - 3, y - 1)$.

8. What is the angle of rotation from A to C about the origin (degree & direction)?

For the rotation:

• Coordinates: $A(1, 1) \to C(-1, -1)$.
• $A \to C$ represents a 180° rotation about the origin (clockwise or counterclockwise).

9. Reflect $\Delta XYZ$ across the x-axis.

Given vertices:

• $X(1, 3)$, $Y(2, 2)$, $Z(3, 3)$.
• Reflection rule across the $x$-axis: $(x, y) \to (x, -y)$.
• Reflected vertices:
  • $X' = (1, -3)$,
  • $Y' = (2, -2)$,
  • $Z' = (3, -3)$.

10. Translate $\Delta XYZ$ using $(x, y) \to (x + 2, y - 1)$.

Given vertices:

• $X(-3, 1)$, $Y(-1, 3)$, $Z(2, 2)$.
• Apply the translation rule:
  • $X' = (-3 + 2, 1 - 1) = (-1, 0)$,
  • $Y' = (-1 + 2, 3 - 1) = (1, 2)$,
  • $Z' = (2 + 2, 2 - 1) = (4, 1)$.

Translated vertices:

• $X' = (-1, 0)$, $Y' = (1, 2)$, $Z' = (4, 1)$.

11. Determine dilation type and scale factor.

For the dilation:

• Original triangle dimensions: $2 \, \text{in}$ (base).
• Dilated triangle dimensions: $5 \, \text{in}$ (base).
• The dilation is an enlargement, and the scale factor is: $\text{Scale Factor} = \frac{5}{2} = 2.5$

Would you like detailed step-by-step explanations for any of these solutions?

Follow-Up Questions

1. How is a line of reflection determined on a coordinate plane?
2. How do you apply translation rules to a triangle’s vertices?
3. What are other ways to verify a 180° rotation about the origin?
4. How does reflection over the x-axis affect the coordinates of a point?
5. What is the difference between enlargement and reduction in dilation?

Tip

When reflecting across a line, always ensure the reflected points are equidistant from the line of reflection!