1. **Problem Statement:** We have triangle $\triangle ABC$ and its dilation $\triangle A'B'C'$ about point $P$ with scale factor 2. We need to determine if the following claims are true or false: - $\overline{AB}$ and $\overline{A'B'}$ lie on the same line. - The perimeters of $\triangle ABC$ and $\triangle A'B'C'$ are the same. 2. **Recall the dilation properties:** - A dilation about a point $P$ with scale factor $k$ maps each point $X$ to $X'$ such that $P$, $X$, and $X'$ are collinear and $PX' = k \cdot PX$. - Dilation preserves the shape but changes the size by the scale factor. - Corresponding segments in the image are parallel to the original segments. - Lengths in the image are multiplied by the scale factor. 3. **Analyze the first claim:** - Since $A'$ and $B'$ are images of $A$ and $B$ under dilation about $P$, points $A$, $A'$, and $P$ are collinear, and points $B$, $B'$, and $P$ are collinear. - However, $\overline{AB}$ and $\overline{A'B'}$ are not necessarily on the same line because dilation moves points along lines through $P$, not necessarily preserving the line $AB$. - But dilation preserves parallelism, so $\overline{A'B'}$ is parallel to $\overline{AB}$. - Therefore, $\overline{AB}$ and $\overline{A'B'}$ are parallel but not the same line. **Conclusion:** The claim "$\overline{AB}$ and $\overline{A'B'}$ are on the same line" is **False**. 4. **Analyze the second claim:** - The perimeter of $\triangle ABC$ is $P = AB + BC + CA$. - After dilation by scale factor 2, each side length doubles: $A'B' = 2 \times AB$, $B'C' = 2 \times BC$, $C'A' = 2 \times CA$. - Therefore, the perimeter of $\triangle A'B'C'$ is $P' = 2AB + 2BC + 2CA = 2P$. **Conclusion:** The claim "The perimeters of $\triangle ABC$ and $\triangle A'B'C'$ are the same" is **False**. **Final answers:** - $\overline{AB}$ and $\overline{A'B'}$ are on the same line: **False** - The perimeters of $\triangle ABC$ and $\triangle A'B'C'$ are the same: **False**