Math Problem Statement

Find the image after the given transformation: Reflection across y = x.

Solution

The problem in the image asks to reflect a shape across the line y=xy = x, and find the coordinates of the new points after the reflection.

Reflection across the line y=xy = x:

When reflecting a point (x,y)(x, y) across the line y=xy = x, the coordinates of the reflected point become (y,x)(y, x). In other words, the x- and y-coordinates swap places.

Let's find the new coordinates of points PP, QQ, HH, and LL after the reflection.

  • Point P has coordinates P(x1,y1)P(x_1, y_1).
  • Point Q has coordinates Q(x2,y2)Q(x_2, y_2).
  • Point H has coordinates H(x3,y3)H(x_3, y_3).
  • Point L has coordinates L(x4,y4)L(x_4, y_4).

After the reflection, the new points PP', QQ', HH', and LL' will be:

  • P=(y1,x1)P' = (y_1, x_1)
  • Q=(y2,x2)Q' = (y_2, x_2)
  • H=(y3,x3)H' = (y_3, x_3)
  • L=(y4,x4)L' = (y_4, x_4)

Let me extract the exact coordinates of the points from the image and compute their reflections.The image shows the points PP, QQ, HH, and LL on the coordinate plane. I'll now determine the coordinates of these points visually and then calculate their reflections across the line y=xy = x.

Coordinates from the image:

  1. P is at (4,3)(-4, -3).
  2. Q is at (2,0)(-2, 0).
  3. H is at (2,3)(-2, 3).
  4. L is at (1,0)(1, 0).

Reflections across y=xy = x:

  1. P': Reflect (4,3)(-4, -3) to get (3,4)(-3, -4).
  2. Q': Reflect (2,0)(-2, 0) to get (0,2)(0, -2).
  3. H': Reflect (2,3)(-2, 3) to get (3,2)(3, -2).
  4. L': Reflect (1,0)(1, 0) to get (0,1)(0, 1).

Final coordinates after reflection:

  • P=(3,4)P' = (-3, -4)
  • Q=(0,2)Q' = (0, -2)
  • H=(3,2)H' = (3, -2)
  • L=(0,1)L' = (0, 1)

Would you like a more detailed explanation on the reflection process or the next steps? Here are five related questions to explore further:

  1. How would the transformation change if reflected across the yy-axis instead?
  2. What are the properties of a reflection across any arbitrary line y=mx+cy = mx + c?
  3. How can we describe a reflection algebraically in terms of matrix transformations?
  4. What happens to the area and orientation of a shape after a reflection?
  5. How do multiple reflections (e.g., over y=xy = x and xx-axis) combine?

Tip: Always check the orientation of points before and after reflection, especially in cases involving complex figures or transformations.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Reflections
Geometric Transformations

Formulas

Reflection across y = x: (x, y) -> (y, x)

Theorems

Theorem of Reflection

Suitable Grade Level

Grades 8-10