Question

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

image of triangle abc with point p and grid

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

$\overline{ab}$ and $\overline{ab}$ are on the same line. true/false

$\overline{ac}$ and $\overline{ac}$ are on distinct parallel lines. true/false

Explanation:

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

Step1: Recall dilation properties

Dilation about a point \( P \) preserves the direction of lines from \( P \). For points \( A, B \) and their images \( A', B' \) under dilation about \( P \), the lines \( PA, PB \) are extended (or contracted) by the scale factor. So, \( A, P, A' \) are colinear and \( B, P, B' \) are colinear. The line \( AB \) and \( A'B' \) should be parallel (or coincident) because dilation preserves the slope (direction) of segments. But since \( A, P, A' \) and \( B, P, B' \) are colinear, the lines \( AB \) and \( A'B' \) are actually the same line (or parallel, but in this case, since \( P \) is a center, the lines through \( AB \) and \( A'B' \) pass through \( P \) and have the same direction, so they are the same line).

Step2: Conclusion

So the claim that \( \overline{AB} \) and \( \overline{A'B'} \) are on the same line is True.

For the second claim: $\boldsymbol{\overline{AC}}$ and $\boldsymbol{\overline{A'C'}}$ are on distinct parallel lines.

Step1: Recall dilation properties

Dilation about a point \( P \) means that \( A, P, A' \) are colinear and \( C, P, C' \) are colinear. The segment \( \overline{AC} \) and its image \( \overline{A'C'} \) under dilation about \( P \) should be such that \( \overline{AC} \parallel \overline{A'C'} \) (because dilation preserves the slope of segments). But since \( A, P, A' \) and \( C, P, C' \) are colinear, the lines \( \overline{AC} \) and \( \overline{A'C'} \) are actually the same line (not distinct parallel lines). So the claim that they are on distinct parallel lines is False.

Answer:

s:

• \( \overline{AB} \) and \( \overline{A'B'} \) are on the same line: True
• \( \overline{AC} \) and \( \overline{A'C'} \) are on distinct parallel lines: False