The problem involves understanding the dilation of a triangle on a coordinate plane and analyzing the properties of the original and dilated triangles.

Given:

• Triangle $\Delta A'B'C'$ is the result of dilating $\Delta ABC$ about point $P$ with a scale factor of 3.
• The task is to determine whether certain claims about the properties of $\Delta ABC$ and $\Delta A'B'C'$ are true or false.

Claims to Analyze:

1. $AB = A'B'$ after dilation.
2. $\angle ABC = \angle A'B'C'$ after dilation.

Analyzing the Claims:

1. Claim: $AB = A'B'$ is False.**

   • A dilation with a scale factor other than 1 changes the lengths of the sides of the triangle. Since the scale factor is 3, every side of $\Delta A'B'C'$ will be 3 times longer than the corresponding side in $\Delta ABC$. Therefore, $AB$ is not equal to $A'B'$.

2. Claim: $\angle ABC = \angle A'B'C'$ is True.**

   • Dilation preserves the angles of the original figure. Therefore, the angles of $\Delta ABC$ will be equal to the corresponding angles in $\Delta A'B'C'$.

Conclusion:

• The first claim $AB = A'B'$ is False.
• The second claim $\angle ABC = \angle A'B'C'$ is True.

Would you like more details, or do you have any questions?

Relative Questions:

1. How does the location of the point $P$ affect the dilation?
2. What happens to the area of the triangle $\Delta A'B'C'$ compared to $\Delta ABC$ when the scale factor is 3?
3. If the scale factor were 1, what would happen to the sides and angles of the triangle after dilation?
4. How would a negative scale factor affect the triangle?
5. Can dilation result in a congruent triangle under certain conditions?

Tip: Always remember that dilation changes the size of a figure but preserves the shape, meaning angles remain the same while side lengths are scaled by the dilation factor.