Math Problem Statement

The trapezoid W'X'Y'Z' is a dilation of the trapezoid WXYZ. What is the scale factor of the dilation?

Solution

To determine the scale factor of the dilation between the trapezoids WXYZWXYZ and WXYZW'X'Y'Z', we need to compare the distances between corresponding points or sides.

Step-by-step process:

  1. Identify corresponding points:

    • The corresponding points of the two trapezoids are WWW \leftrightarrow W', XXX \leftrightarrow X', YYY \leftrightarrow Y', and ZZZ \leftrightarrow Z'.
  2. Measure distance between corresponding points:

    • Let's find the coordinates of the vertices of trapezoid WXYZWXYZ (larger trapezoid):
      • W(8,2)W(8, -2), X(8,6)X(8, 6), Y(6,6)Y(-6, 6), Z(6,8)Z(-6, -8).
    • Now the coordinates of the corresponding points of the dilated trapezoid WXYZW'X'Y'Z':
      • W(4,1)W'(4, -1), X(4,3)X'(4, 3), Y(2,3)Y'(-2, 3), Z(2,4)Z'(-2, -4).
  3. Calculate distances:

    • For simplicity, we will measure the distance between WW and WW', as well as between ZZ and ZZ', to find the scale factor.
    • Distance from WW to the origin: W(8,2)W(8, -2), so its distance from the origin is 82+(2)2=64+4=68\sqrt{8^2 + (-2)^2} = \sqrt{64 + 4} = \sqrt{68}.
    • Distance from WW' to the origin: W(4,1)W'(4, -1), so its distance from the origin is 42+(1)2=16+1=17\sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17}.
  4. Find scale factor:

    • The scale factor is the ratio of the distances from the center of dilation (which we assume is the origin): Scale factor=Distance from origin to WDistance from origin to W=1768=12.\text{Scale factor} = \frac{\text{Distance from origin to } W'}{\text{Distance from origin to } W} = \frac{\sqrt{17}}{\sqrt{68}} = \frac{1}{2}.

Thus, the scale factor of the dilation is 12\boxed{\frac{1}{2}}.

Would you like more details or have any questions?


Here are 5 related questions for further exploration:

  1. How would the trapezoid's area change if the scale factor was 3 instead of 12\frac{1}{2}?
  2. What is the general formula for the scale factor in dilations centered at the origin?
  3. How can you determine the center of dilation if it's not given?
  4. How does the perimeter of a figure change after dilation?
  5. Can the scale factor be negative, and what would that imply geometrically?

Tip: When finding the scale factor in dilation, always ensure to use the distance between corresponding points, and make sure the center of dilation is clear (often the origin).

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Math Problem Analysis

Mathematical Concepts

Geometry
Dilation
Scale Factor

Formulas

Scale factor = Distance between corresponding points in dilated and original figure / Original distance

Theorems

Dilation Theorem

Suitable Grade Level

Grades 8-10