Let's start by analyzing the given problem carefully and determining the truth of each claim about the properties of the triangles $\triangle ABC$ and $\triangle A'B'C'$ based on the dilation transformation.

Step 1: Coordinates of the Points

• The coordinates of points $A$, $B$, and $C$ for $\triangle ABC$ are:

  • $A(2, 3)$
  • $B(8, 3)$
  • $C(4, 7)$

• The midpoint $P$ of side $BC$ has coordinates $P(6, 5)$. Point $P$ is the center of dilation.

Step 2: Understanding the Dilation

The dilation of $\triangle ABC$ about point $P(6, 5)$ by a scale factor of $2$ transforms each point of the triangle by doubling the distance from the center of dilation $P$.

To find the coordinates of $A'$, $B'$, and $C'$, we apply the dilation transformation:

• For point $A(2, 3)$, the vector from $P(6, 5)$ to $A(2, 3)$ is $\langle 2 - 6, 3 - 5 \rangle = \langle -4, -2 \rangle$. Doubling the vector, we get $\langle -8, -4 \rangle$. Adding this to the coordinates of $P$, we get: $A' = (6, 5) + (-8, -4) = (-2, 1)$

• For point $B(8, 3)$, the vector from $P(6, 5)$ to $B(8, 3)$ is $\langle 8 - 6, 3 - 5 \rangle = \langle 2, -2 \rangle$. Doubling the vector, we get $\langle 4, -4 \rangle$. Adding this to the coordinates of $P$, we get: $B' = (6, 5) + (4, -4) = (10, 1)$

• For point $C(4, 7)$, the vector from $P(6, 5)$ to $C(4, 7)$ is $\langle 4 - 6, 7 - 5 \rangle = \langle -2, 2 \rangle$. Doubling the vector, we get $\langle -4, 4 \rangle$. Adding this to the coordinates of $P$, we get: $C' = (6, 5) + (-4, 4) = (2, 9)$

Thus, the coordinates of the dilated points are:

• $A'(-2, 1)$
• $B'(10, 1)$
• $C'(2, 9)$

Step 3: Analyzing the Claims

1. Claim 1: $\overline{AB}$ and $\overline{A'B'}$ are on the same line.

• False. Points $A(2, 3)$, $B(8, 3)$, $A'(-2, 1)$, and $B'(10, 1)$ do not lie on the same straight line.
  • The line $AB$ is horizontal, as both $A$ and $B$ have the same $y$-coordinate ($y = 3$).
  • The line $A'B'$ is not horizontal, as the $y$-coordinates of $A'$ and $B'$ are different ($y = 1$ for both), but the line passing through them does not share the same orientation as $\overline{AB}$.

2. Claim 2: The perimeters of $\triangle ABC$ and $\triangle A'B'C'$ are the same.

• False. The perimeter of a dilated triangle is proportional to the perimeter of the original triangle by the scale factor. Since the scale factor is 2, the perimeter of $\triangle A'B'C'$ will be twice that of $\triangle ABC$. Therefore, the perimeters are not the same.

Conclusion:

• The first claim is false.
• The second claim is false.

Further Questions:

1. How do we calculate the perimeter of a triangle given its vertices?
2. What happens to the area of a triangle under dilation?
3. How does the center of dilation affect the shape of a triangle during transformation?
4. If the dilation scale factor was different, how would that affect the perimeters of the triangles?
5. How do we check if two points lie on the same line geometrically?

Tip: When checking if two segments lie on the same line, you can compute the slope of the line between the points and check if the slopes are equal. If they are, the points are collinear.