Tavon used a graphing program to graph a circle centered at the origin with a radius of 3 in the coordinate plane. He then applied two transformations to the original circle; first a dilation and then a translation. The new circle has a radius of 6 centered at the point (−1,0). Which of the following statements best explains the relationship between the original circle and its image under the sequence of transformations?ResponsesThe circle and its image are congruent because the sequence of transformations included a dilation with a scale factor of 2.Answer A: The circle and its image are congruent because the sequence of transformations included a dilation with a scale factor of 2 .AThe circle and its image are congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.Answer B: The circle and its image are congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.BThe circle and its image are similar but not congruent because the sequence of transformations included a dilation with a scale factor of 2.Answer C: The circle and its image are similar but not congruent because the sequence of transformations included a dilation with a scale factor of 2 .CThe circle and its image are similar but not congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.
Question
Tavon used a graphing program to graph a circle centered at the origin with a radius of 3 in the coordinate plane. He then applied two transformations to the original circle; first a dilation and then a translation. The new circle has a radius of 6 centered at the point (−1,0). Which of the following statements best explains the relationship between the original circle and its image under the sequence of transformations?ResponsesThe circle and its image are congruent because the sequence of transformations included a dilation with a scale factor of 2.Answer A: The circle and its image are congruent because the sequence of transformations included a dilation with a scale factor of 2 .AThe circle and its image are congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.Answer B: The circle and its image are congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.BThe circle and its image are similar but not congruent because the sequence of transformations included a dilation with a scale factor of 2.Answer C: The circle and its image are similar but not congruent because the sequence of transformations included a dilation with a scale factor of 2 .CThe circle and its image are similar but not congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.
Solution
To determine the correct response, let's analyze the transformations step by step:
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Original Circle: The original circle is centered at the origin (0,0) with a radius of 3.
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Dilation: The circle undergoes a dilation. The new circle has a radius of 6. Since the radius has doubled (from 3 to 6), the scale factor of the dilation is 2.
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Translation: The new circle is centered at the point (−1,0). This indicates a horizontal translation of 1 unit to the left.
Now, let's evaluate the given responses:
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Answer A: The circle and its image are congruent because the sequence of transformations included a dilation with a scale factor of 2.
- This statement is incorrect because a dilation with a scale factor of 2 changes the size of the circle, making the new circle not congruent to the original.
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Answer B: The circle and its image are congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.
- This statement is incorrect because while the translation does not change the size of the circle, the dilation does. Therefore, the circles are not congruent.
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Answer C: The circle and its image are similar but not congruent because the sequence of transformations included a dilation with a scale factor of 2.
- This statement is correct. The dilation changes the size of the circle, making the new circle similar to the original (same shape but different size) but not congruent.
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Answer D: The circle and its image are similar but not congruent because the sequence of transformations included a horizontal translation of 1 unit to the left.
- This statement is incorrect because the translation alone does not affect the similarity or congruence. The key factor is the dilation.
Therefore, the correct response is:
Answer C: The circle and its image are similar but not congruent because the sequence of transformations included a dilation with a scale factor of 2.
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