The problem asks to transform $\triangle ABC$ into $\triangle A'B'C'$ by applying a translation and a dilation. Here's the step-by-step process to solve this:

Step 1: Identify the Translation

First, we need to figure out the translation rule that maps $\triangle ABC$ to its new location.

• Point $A$ is originally at $(4, 8)$, and point $A'$ is at $(-8, 10)$.
• Point $B$ is at $(2, 2)$, and point $B'$ is at $(-6, -6)$.
• Point $C$ is at $(8, 2)$, and point $C'$ is at $(0, -6)$.

To find the translation rule, we subtract the original coordinates from the new coordinates:

For $A$:

• $x$: $-8 - 4 = -12$
• $y$: $10 - 8 = 2$

So, the translation rule is: $T(x, y) = (x - 12, y + 2)$

Step 2: Find the Scale Factor of the Dilation

Dilation involves scaling the points relative to the origin.

To find the scale factor, we can compute the ratio of corresponding distances from the origin to points before and after dilation.

For point $A$ and $A'$:

• Distance of $A$ from origin = $\sqrt{(4^2 + 8^2)} = \sqrt{16 + 64} = \sqrt{80}$
• Distance of $A'$ from origin = $\sqrt{((-8)^2 + 10^2)} = \sqrt{64 + 100} = \sqrt{164}$

Scale factor $k = \frac{\text{Distance of } A'}{\text{Distance of } A} = \frac{\sqrt{164}}{\sqrt{80}} = \frac{\sqrt{164/80}} = \frac{\sqrt{41}}{2}$.

Hence, the scale factor is approximately $\frac{\sqrt{41}}{2}$ or around $1.6$ (in decimal form).

Final Answer:

• Translation rule: $T(x, y) = (x - 12, y + 2)$
• Scale factor: $\frac{\sqrt{41}}{2}$ or approximately $1.6$.

Would you like any additional clarification on any part of the solution?

Here are five related questions to explore:

1. How does the translation rule affect each point of the triangle?
2. What is the exact scale factor of dilation in terms of radicals?
3. How would the translation rule change if the translation happened after the dilation?
4. Can you confirm the translation rule for point $B$ or $C$?
5. How would a negative scale factor affect the dilation?

Tip: Always verify your transformation by applying both the translation and dilation to each point and checking if it maps to the new location correctly!