Math Problem Statement

Example 5: ΔPQR reflected over the line y = -x. What is the general rule for reflecting across y = -x? Example 6: Reflect ΔEFG over the line x = 1. Example 7: What rule represents the transformation for ΔABC to ΔA'B'C'?

Solution

Let's break down the examples in your image.

Example 5:

  • Reflection of triangle ΔPQR\Delta PQR over the line y=xy = -x.
  • The reflection rule for reflecting over y=xy = -x is to switch the coordinates and negate both: (x,y)(y,x)(x, y) \to (-y, -x).
  • Applying this rule to each point:
    • P(1,2)P(-1, 2) becomes P(2,1)P'(-2, 1)
    • Q(3,1)Q(3, 1) becomes Q(1,3)Q'(-1, -3)
    • R(2,2)R(2, -2) becomes R(2,2)R'(-2, -2)

Example 6:

  • Reflection of triangle ΔEFG\Delta EFG over the line x=1x = 1.
  • To reflect a point over the line x=1x = 1, we calculate the horizontal distance from each point to the line and move it the same distance on the opposite side.
    • For E(4,6)E(-4, 6), the distance to x=1x = 1 is 5 units. The reflected point is E(6,6)E'(6, 6).
    • For F(1,6)F(1, 6), the distance is 0, so FF remains at F(1,6)F'(1, 6).
    • For G(2,2)G(-2, -2), the distance to x=1x = 1 is 3 units. The reflected point is G(4,2)G'(4, -2).

Example 7:

  • Transformation rule for ΔABC\Delta ABC to ΔABC\Delta A'B'C'.
  • We can observe that each point is reflected across the yy-axis, meaning the transformation rule is a reflection over the yy-axis: (x,y)(x,y)(x, y) \to (-x, y).
    • A(4,0)A(4,0)A(4, 0) \to A'(-4, 0)
    • B(7,5)B(7,5)B(-7, -5) \to B'(7, -5)
    • C(2,6)C(2,6)C(2, 6) \to C'(-2, 6)

Would you like more details or have any questions?

Related Questions:

  1. How do we determine the reflection rule for other lines like y=xy = x?
  2. What changes when reflecting across lines other than the axes?
  3. How can we verify if a point has been reflected correctly?
  4. Can we derive a formula for any reflection over arbitrary lines?
  5. How are reflections related to symmetry in geometry?

Tip: Always visualize the reflection by counting the distance from the point to the reflection line!

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Math Problem Analysis

Mathematical Concepts

Geometry
Reflections
Coordinate Plane
Transformations

Formulas

Reflection over y = -x: (x, y) → (-y, -x)
Reflection over x = 1: Calculate distance from each point to x = 1, then move the same distance on the opposite side
Reflection over y-axis: (x, y) → (-x, y)

Theorems

Reflection Theorem: A reflection flips a figure over a line, creating a mirror image on the opposite side of the line.

Suitable Grade Level

Grades 8-10