Question

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

Explanation:

For the first claim: $\boldsymbol{\overline{BC}}$ and $\boldsymbol{\overline{B'C'}}$ are on the same line.

Step1: Recall dilation properties

Dilation about a point \( P \) preserves collinearity. So, points \( B \), \( P \), \( B' \) are collinear, and points \( C \), \( P \), \( C' \) are collinear. Since \( B \), \( P \), \( C \) are on line \( BC \) (from the diagram, \( B \) and \( C \) are vertical, \( P \) is on that vertical line), then \( B' \) and \( C' \) will also be on that line after dilation (scale factor 2 about \( P \)).

Step2: Conclude collinearity

Thus, \( \overline{BC} \) and \( \overline{B'C'} \) lie on the same vertical line (through \( P \), \( B \), \( C \)). So the claim is true.

For the second claim: $\boldsymbol{\overline{AB}}$ and $\boldsymbol{\overline{A'B'}}$ are both parallel to the \( \boldsymbol{x} \)-axis.

Step1: Analyze \( \overline{AB} \)

From the diagram, \( A \) and \( B \) have the same \( y \)-coordinate (both \( y = 1 \)), so \( \overline{AB} \) is horizontal, hence parallel to the \( x \)-axis.

Step2: Analyze \( \overline{A'B'} \) after dilation

Dilation about \( P \) preserves the direction (slope) of segments. Since \( \overline{AB} \) is horizontal, \( \overline{A'B'} \) (image of \( \overline{AB} \) under dilation about \( P \)) will also be horizontal (same slope, 0), so it is parallel to the \( x \)-axis. Thus, the claim is true.

Answer:

s:

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