Math Problem Statement

Draw each of the images described in the steps below. Step 1: First, starting with the given figure, perform a dilation centered at the origin with a scale factor of 1/3. Step 2: Then from the resulting figure, perform a dilation centered at the origin with a scale factor of 2. Fill in the blank: Performing Steps 1 and 2 is equivalent to a single dilation centered at the origin with a scale factor of __.

Solution

To address the problem:

Part (a): Steps of Dilations

  1. Step 1: Dilate the original figure by a scale factor of 13\frac{1}{3}:

    • Each vertex of the triangle is moved to a position that is 13\frac{1}{3} the distance from the origin (center of dilation).
  2. Step 2: Dilate the resulting figure by a scale factor of 22:

    • The new figure from Step 1 is again dilated, moving each point to a position 22 times its distance from the origin.

I can create a diagram to visually represent these transformations. Let me know if you'd like that!


Part (b): Equivalent Scale Factor

The combined effect of the two dilations can be found by multiplying the two scale factors together: Combined Scale Factor=13×2=23.\text{Combined Scale Factor} = \frac{1}{3} \times 2 = \frac{2}{3}.

Thus, performing Steps 1 and 2 is equivalent to a single dilation centered at the origin with a scale factor of 23\frac{2}{3}.


Would you like me to visualize or calculate the transformations further? Here are 5 related questions you might want to explore:

  1. How would you plot the vertices after each transformation step?
  2. What is the general rule for combining multiple dilations?
  3. How does the choice of center of dilation affect the transformation?
  4. Can this method be applied to other geometric figures beyond triangles?
  5. How would the process change if the dilation centers were not the origin?

Tip: When combining multiple transformations, consider their cumulative effect by multiplying their respective transformation factors.

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Math Problem Analysis

Mathematical Concepts

Geometry
Dilations
Scale Factor
Transformations

Formulas

Combined scale factor = product of individual scale factors

Theorems

Properties of Dilations

Suitable Grade Level

Grades 8-10