In a dilation, each point of a shape moves away from or towards a center point (in this case, the origin) based on a specified scale factor.
For triangle ABC, when we have a dilation with a scale factor of 5 and the center of dilation at the origin (0, 0), the following happens:
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Each vertex of triangle ABC is multiplied by the scale factor (5). This means that if the coordinates of a vertex are (x, y), the new coordinates for the corresponding vertex A′ of triangle A′B′C′ will be (5x, 5y).
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The triangle A′B′C′ will not only be 5 times larger than triangle ABC but will also be positioned further away from the origin. Each vertex that was previously at distance d from the origin will now be at distance 5d from the origin.
Based on these principles, let's analyze the statements:
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Triangle A′B′C′ is 5 times smaller than triangle ABC and is 5 times closer to the center point of dilation.
- This is incorrect; dilation with a scale factor of 5 makes the new triangle larger, not smaller.
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Triangle A′B′C′ is 5 times smaller than triangle ABC and is 5 times as far from the center point of dilation.
- This is also incorrect; as explained, the triangle is larger, not smaller.
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Triangle A′B′C′ is 5 times as large as triangle ABC and is 5 times closer to the center point of dilation.
- This is incorrect; the triangle is larger, but it is actually farther from the center, not closer.
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Triangle A′B′C′ is 5 times as large as triangle ABC and is 5 times as far from the center point of dilation.
- This statement is correct; the new triangle is 5 times larger and is positioned 5 times farther away from the center of dilation at the origin.
Thus, the correct option is 4. Triangle A′B′C′ is 5 times as large as triangle ABC and is 5 times as far from the center point of dilation.