Math Problem Statement

Point O is the center of dilation. Triangle X Y Z is dilated to create triangle X prime Y prime Z prime. The length of O Z is 2 and the length of Z Z prime is 4. The length of side X prime Y prime is 9 and the length of side Y prime Z prime is 12. Triangle XYZ was transformed by DO,3. What is the length of side XY?

1.5 4.5 3 18

Solution

We are given that triangle XYZXYZ is dilated by a scale factor of 3 (DO,3D_{O,3}), and we need to find the length of side XYXY of the pre-image triangle before dilation.

Information provided:

  • The scale factor k=3k = 3 (since it was dilated by DO,3D_{O,3}).
  • The length of side XYX'Y' (in the dilated triangle) is 9.

Since the scale factor of dilation is 3, the sides of the original triangle XYZXYZ are 13\frac{1}{3} of the corresponding sides of the dilated triangle XYZX'Y'Z'.

Calculation:

To find the length of side XYXY, we divide the length of the corresponding side XYX'Y' by the scale factor:

XY=XYk=93=3XY = \frac{X'Y'}{k} = \frac{9}{3} = 3

Thus, the length of side XYXY is 3.

Answer:

The correct option is 3.

Would you like more details or explanations?

Here are 5 related questions:

  1. What is the general formula for finding a side length after dilation?
  2. How do you determine the center of dilation from coordinates?
  3. Can the scale factor be negative, and what does that imply geometrically?
  4. If the scale factor was 2, what would the length of side XYXY be?
  5. What happens to the angles of a triangle when it undergoes dilation?

Tip: When dealing with dilation, remember that lengths change by the scale factor, but angles remain the same.

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Math Problem Analysis

Mathematical Concepts

Geometry
Dilation
Scale Factor

Formulas

Side length of pre-image = Side length of image / Scale factor

Theorems

Dilation Theorem: A dilation maps a figure to a similar figure, maintaining angle measures and altering side lengths by a constant scale factor

Suitable Grade Level

Grades 8-10