Question

Quadrilateral ABCD is dilated by a scale factor of two thirds centered around (1, 2). Quadrilateral ABCD is shown on a coordinate plane with A at (1, 2), B at (2, 3), C at (4, 2), and D at (2, 1). Which statement is true about the dilation?

1) Segment B' D' will run through (1, 2) and will be shorter than segment BD.
2) Segment B' D' will run through (1, 2) and will be longer than segment BD.
3) Segment B' D' will be parallel to segment BD and will be shorter than segment BD.
4) Segment B' D' will be parallel to segment BD and will be longer than segment BD.

Answer 1

The correct statement about the dilation of quadrilateral ABCD by a factor of two-thirds with the center of dilation at (1, 2) is that the resulting segment B'D' will be parallel to the original segment BD and will also be shorter than BD.

Step-by-step explanation:

When a shape is dilated by a scale factor from a center of dilation, each point on the shape is moved along a line through the point and the center of dilation to a distance proportional to the original distance from the center. In the given question, quadrilateral ABCD is dilated by a scale factor of two-thirds from the center at (1, 2). Since point A is at the center of dilation, it will not move, while points B, C, and D will move closer to (1, 2).

Specifically, segment BD will be transformed into segment B'D' through the dilation. The points B and D will move along lines that intersect at the dilation's center (1, 2), so segment B'D' will also pass through this center. Because the scale factor is two-thirds, which is less than 1, B'D' will be shorter than BD. Additionally, segment B'D' will be parallel to segment BD since dilation preserves angles and parallelism.

So, the correct statement about the dilation is that segment B'D' will be parallel to segment BD and will be shorter than segment BD. That aligns with the third provided statement in the question.