Question
Describe the dilation of triangle ABC , with a scale factor of 5 and a center point of dilation at the origin (0, 0) .
1. Triangle A′B′C′ is 5 times smaller than triangle ABC and is 5 times closer to the center point of dilation.
2. Triangle A′B′C′ is 5 times smaller than triangle ABC and is 5 times as far from the center point of dilation.
3. Triangle A′B′C′ is 5 times as large as triangle ABC and is 5 times closer to the center point of dilation.
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.
1. Triangle A′B′C′ is 5 times smaller than triangle ABC and is 5 times closer to the center point of dilation.
2. Triangle A′B′C′ is 5 times smaller than triangle ABC and is 5 times as far from the center point of dilation.
3. Triangle A′B′C′ is 5 times as large as triangle ABC and is 5 times closer to the center point of dilation.
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.
Answers
GPT-4o mini
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:
1. 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).
2. 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:
1. **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.
2. **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.
3. **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.
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.**
- 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.**
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:
1. 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).
2. 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:
1. **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.
2. **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.
3. **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.
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.**
- 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.**