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 over the line .
- The reflection rule for reflecting over is to switch the coordinates and negate both: .
- Applying this rule to each point:
- becomes
- becomes
- becomes
Example 6:
- Reflection of triangle over the line .
- To reflect a point over the line , we calculate the horizontal distance from each point to the line and move it the same distance on the opposite side.
- For , the distance to is 5 units. The reflected point is .
- For , the distance is 0, so remains at .
- For , the distance to is 3 units. The reflected point is .
Example 7:
- Transformation rule for to .
- We can observe that each point is reflected across the -axis, meaning the transformation rule is a reflection over the -axis: .
Would you like more details or have any questions?
Related Questions:
- How do we determine the reflection rule for other lines like ?
- What changes when reflecting across lines other than the axes?
- How can we verify if a point has been reflected correctly?
- Can we derive a formula for any reflection over arbitrary lines?
- 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