To solve this translation problem, let's proceed step by step:

Step 1: Determine the translation vector

We are told that the point $A(-3, -2)$ is translated to $A'(-2, 0)$. To find the translation vector, calculate the change in the $x$-coordinate and $y$-coordinate:

$\text{Translation vector} = (x_2 - x_1, y_2 - y_1)$

Here:

• $x_1 = -3$, $y_1 = -2$
• $x_2 = -2$, $y_2 = 0$

$\text{Translation vector} = (-2 - (-3), 0 - (-2)) = (1, 2)$

Thus, the translation vector is $(1, 2)$.

Step 2: Apply the translation to $B(-6, -4)$

To find the coordinates of $B'$, add the translation vector $(1, 2)$ to $B$:

$B' = (x_B + 1, y_B + 2)$

Here:

• $x_B = -6$, $y_B = -4$

$B' = (-6 + 1, -4 + 2) = (-5, -2)$

So, the coordinates of $B'$ are $(-5, -2)$.

Step 3: Apply the translation to $C(1, 2)$

To find the coordinates of $C'$, add the translation vector $(1, 2)$ to $C$:

$C' = (x_C + 1, y_C + 2)$

Here:

• $x_C = 1$, $y_C = 2$

$C' = (1 + 1, 2 + 2) = (2, 4)$

So, the coordinates of $C'$ are $(2, 4)$.

Final Answer:

The coordinates of the translated points are:

• $B'(-5, -2)$
• $C'(2, 4)$

Do you want a more detailed explanation of any step or have further questions?

Here are 5 related questions to expand your understanding:

1. How do you calculate a translation vector between two points?
2. If a point $D(4, -5)$ is translated by the vector $(3, 6)$, what are its new coordinates?
3. What happens to the coordinates if the translation vector is $(0, 0)$?
4. How can translations be represented using vector addition?
5. What is the difference between translation, rotation, and reflection in geometry?

Tip: Always double-check the translation vector by verifying changes in both coordinates!