Question
select all the statement that must be true for any scaled copy Q of Polygon P
Answers
GPT-4o mini
When discussing scaled copies of a polygon, there are several properties that must hold true. Here are statements that generally must be true for any scaled copy \( Q \) of a polygon \( P \):
1. **Shape Preservation**: The shape of polygon \( Q \) must be the same as the shape of polygon \( P \). This means that the angles of \( Q \) must be congruent to the corresponding angles of \( P \).
2. **Proportional Sides**: The sides of polygon \( Q \) must be proportional to the sides of polygon \( P \). This means that for any two corresponding sides \( a \) from \( P \) and \( b \) from \( Q \), the ratio \( \frac{b}{a} \) is constant (the scale factor) across all sides.
3. **Center of Scaling**: If polygon \( P \) is scaled about a specific point (the center of scaling), then every point in polygon \( Q \) will also be positioned relative to the center of scaling in the same way as the corresponding points in polygon \( P \).
4. **Parallel Corresponding Sides**: The corresponding sides of \( P \) and \( Q \) must be parallel. This means that if two sides of \( P \) are parallel, the corresponding sides of \( Q \) will also be parallel.
5. **Vertices Correspondence**: Each vertex of polygon \( Q \) corresponds to a vertex of polygon \( P \) and maintains a relationship defined by the scale factor.
When applying these properties, it's important to remember that they apply regardless of whether \( Q \) is a reduction (scale factor less than 1) or an enlargement (scale factor greater than 1) of \( P \).
1. **Shape Preservation**: The shape of polygon \( Q \) must be the same as the shape of polygon \( P \). This means that the angles of \( Q \) must be congruent to the corresponding angles of \( P \).
2. **Proportional Sides**: The sides of polygon \( Q \) must be proportional to the sides of polygon \( P \). This means that for any two corresponding sides \( a \) from \( P \) and \( b \) from \( Q \), the ratio \( \frac{b}{a} \) is constant (the scale factor) across all sides.
3. **Center of Scaling**: If polygon \( P \) is scaled about a specific point (the center of scaling), then every point in polygon \( Q \) will also be positioned relative to the center of scaling in the same way as the corresponding points in polygon \( P \).
4. **Parallel Corresponding Sides**: The corresponding sides of \( P \) and \( Q \) must be parallel. This means that if two sides of \( P \) are parallel, the corresponding sides of \( Q \) will also be parallel.
5. **Vertices Correspondence**: Each vertex of polygon \( Q \) corresponds to a vertex of polygon \( P \) and maintains a relationship defined by the scale factor.
When applying these properties, it's important to remember that they apply regardless of whether \( Q \) is a reduction (scale factor less than 1) or an enlargement (scale factor greater than 1) of \( P \).