Question

triangle △abc is the result of dilating △abc about point p by a scale factor of 2.
determine whether each claim about the properties of △abc and △abc is true or false.
\\(\overline{bc}\\) and \\(\overline{bc}\\) are on the same line. \ttrue/false
\\(\overline{ab}\\) and \\(\overline{ab}\\) are both parallel to the \\(x\\)-axis. \ttrue/false

Explanation:

Step1: Analyze \( \overline{BC} \) and \( \overline{B'C'} \)

Dilation about a point \( P \) preserves the line through the center of dilation and the corresponding points. Since \( \triangle A'B'C' \) is a dilation of \( \triangle ABC \) about \( P \), points \( B \), \( P \), \( B' \) are colinear and \( C \), \( P \), \( C' \) are colinear. So \( \overline{BC} \) and \( \overline{B'C'} \) lie on the same line (through \( P \)). Thus, the first claim is True.

Step2: Analyze \( \overline{AB} \) and \( \overline{A'B'} \)

Dilation preserves parallelism. \( \overline{AB} \) is parallel to the \( x \)-axis (horizontal line). After dilation about \( P \), \( \overline{A'B'} \) will be parallel to \( \overline{AB} \), so it is also parallel to the \( x \)-axis. Thus, the second claim is True.

Answer:

For "\( \overline{BC} \) and \( \overline{B'C'} \) are on the same line": True
For "\( \overline{AB} \) and \( \overline{A'B'} \) are both parallel to the \( x \)-axis": True