In a dilation of a geometric figure, what happens to the shape and size of the figure?
A. The shape becomes larger, and the size remains the same.
B. The shape remains the same, and the size changes.
C. The shape and size both change.
D. The shape and size remain the same.
The Correct answer and Explanation is:
Correct Answer:
B. The shape remains the same, and the size changes.
Explanation:
A dilation in geometry is a transformation that alters the size of a figure while preserving its shape. This process involves scaling the figure either up (enlargement) or down (reduction) based on a specified scale factor. The critical characteristic of a dilation is that it maintains the original proportions and angles of the figure, ensuring that the overall shape remains unchanged.
When a figure is dilated, each point of the figure moves away from (or towards) a fixed point known as the center of dilation. The distance from each point to the center of dilation is multiplied by the scale factor. If the scale factor is greater than one, the figure enlarges; if it is between zero and one, the figure reduces in size. Importantly, although the dimensions of the figure change, the angles and the relative positions of the points remain constant, preserving the similarity of the shape.
For example, consider a triangle with vertices at points A, B, and C. If the triangle undergoes a dilation with a scale factor of 2 centered at point O, each vertex (A, B, C) will be moved to a new position (A’, B’, C’) such that OA’ = 2 * OA, OB’ = 2 * OB, and OC’ = 2 * OC. Although the lengths of the sides of the triangle change, the angles formed at each vertex remain the same, demonstrating that the shape is preserved.
In conclusion, a dilation results in a change in size while the shape of the figure remains identical, validating that the correct answer to the question is B.