Math Problem Statement

The figure below is dilated by a factor of 1/2 centered at the origin. Plot the resulting image.

Solution

The problem requires plotting the resulting image of the figure after applying a dilation by a factor of 12\frac{1}{2} centered at the origin. Let's work through the solution step by step.

Steps to Solve:

  1. Understand Dilation:

    • A dilation transformation scales the coordinates of each vertex of the figure by the dilation factor kk, relative to the center of dilation (in this case, the origin).
    • Formula: (x,y)(kx,ky)(x, y) \to (kx, ky).
  2. Apply Dilation Factor 12\frac{1}{2}:

    • Each vertex of the original figure will have its xx and yy-coordinates multiplied by 12\frac{1}{2}.
  3. Original Coordinates of Vertices: From the image, approximate coordinates of the vertices are:

    • C(4,8)C(-4, -8)
    • D(6,6)D(6, -6)
    • E(4,8)E(4, 8)
    • F(6,4)F(-6, 4)
  4. Calculate New Coordinates: Using k=12k = \frac{1}{2}:

    • C=(412,812)=(2,4)C' = (-4 \cdot \frac{1}{2}, -8 \cdot \frac{1}{2}) = (-2, -4)
    • D=(612,612)=(3,3)D' = (6 \cdot \frac{1}{2}, -6 \cdot \frac{1}{2}) = (3, -3)
    • E=(412,812)=(2,4)E' = (4 \cdot \frac{1}{2}, 8 \cdot \frac{1}{2}) = (2, 4)
    • F=(612,412)=(3,2)F' = (-6 \cdot \frac{1}{2}, 4 \cdot \frac{1}{2}) = (-3, 2)
  5. Plot the New Image:

    • Draw the quadrilateral connecting the points C(2,4)C'(-2, -4), D(3,3)D'(3, -3), E(2,4)E'(2, 4), and F(3,2)F'(-3, 2).

Would you like me to sketch the resulting transformation for you? Let me know if you want additional explanations or further assistance!


Related Questions:

  1. What happens to the area of a figure when it is dilated by a factor of 12\frac{1}{2}?
  2. How would the coordinates change if the dilation factor were 22 instead of 12\frac{1}{2}?
  3. What are some real-world applications of dilation transformations?
  4. How does the center of dilation impact the transformed figure?
  5. Can a dilation result in a figure with overlapping or negative coordinates? Why or why not?

Tip:

Always double-check your vertex coordinates before applying transformations to ensure accuracy.

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Math Problem Analysis

Mathematical Concepts

Dilation
Transformations
Coordinate Geometry

Formulas

(x, y) → (kx, ky)

Theorems

Basic properties of dilation

Suitable Grade Level

Grades 8-10