Question

triangle $\triangle abc$ is the result of dilating $\triangle abc$ about point $p$ by a scale factor of 3.

image of triangle abc and point p on a grid

determine whether each claim about the properties of $\triangle abc$ and $\triangle abc$ is true or false.

$ab$ and $ab$ are on the same line. \hfill $\boxed{\text{true}}$ (dropdown)

$ac$ and $ac$ are on distinct parallel lines. \hfill $\boxed{\text{true/false}}$ (dropdown)

Explanation:

Step1: Analyze AB and A'B'

When dilating about point P, the line connecting a vertex and its image passes through P. For segment AB, the line AB does not pass through P, but the image A'B' lies on the line that extends from P through A and P through B. However, AB and A'B' are collinear (lie on the same line) because dilation preserves the line through the center and the point, so the original segment and its image lie on the same infinite line.

Step2: Analyze AC and A'C'

Segment AC passes through the center of dilation P. When a segment passes through the dilation center, its image lies on the same line as the original segment, not a distinct parallel line. Parallel lines only occur when the original segment does not pass through the dilation center.

Answer:

1. AB and A'B' are on the same line: True
2. AC and A'C' are on distinct parallel lines: False