To solve this problem, let's go through each part systematically.

Part (a): Determine the coordinates of $R'$, $S'$, and $T'$

A dilation transformation with a scale factor of $\frac{3}{2}$ means that each coordinate $(x, y)$ will be scaled by multiplying both $x$ and $y$ by $\frac{3}{2}$.

Let's apply this to each point:

1. For $R(6, 4)$: $R' = \left(\frac{3}{2} \times 6, \frac{3}{2} \times 4\right) = (9, 6)$

2. For $S(6, -2)$: $S' = \left(\frac{3}{2} \times 6, \frac{3}{2} \times -2\right) = (9, -3)$

3. For $T(-2, -4)$: $T' = \left(\frac{3}{2} \times -2, \frac{3}{2} \times -4\right) = (-3, -6)$

So the answers for part (a) are:

• $R' = (9, 6)$
• $S' = (9, -3)$
• $T' = (-3, -6)$

Part (b): Choose the correct rule for the dilation mapping

The dilation rule with a scale factor of $\frac{3}{2}$ applied to any point $(x, y)$ is: $(x, y) \rightarrow \left(\frac{3}{2}x, \frac{3}{2}y\right)$

Looking at the options provided, the correct answer for part (b) is:

• $(x, y) \rightarrow \left(\frac{3}{2}x, \frac{3}{2}y\right)$.

Summary

1. Coordinates after dilation:

   • $R' = (9, 6)$
   • $S' = (9, -3)$
   • $T' = (-3, -6)$

2. Dilation rule: $(x, y) \rightarrow \left(\frac{3}{2}x, \frac{3}{2}y\right)$

Would you like further details on dilation transformations?

Related Questions

1. How would the coordinates change if the scale factor was $\frac{1}{2}$ instead?
2. What is the effect of a scale factor less than 1 on the shape of the triangle?
3. How do you find the area of $\triangle R'S'T'$ after dilation?
4. What happens to a shape if the scale factor is negative?
5. How does a dilation differ from a rotation in terms of coordinate changes?

Tip

Remember, in a dilation centered at the origin, each coordinate is simply scaled by the factor, making calculations straightforward.