To solve the given transformation problem and determine the coordinates of $C'$, let's break it into two parts:

1. Translation of B
   The transformation $(x, y) \to (x - 4, y - 2)$ translates a point by subtracting 4 from the x-coordinate and 2 from the y-coordinate. This is applied to point $B$. If $B$'s coordinates are not provided, they must be given or symbolized as $(b_x, b_y)$.

2. Rotation of C by 180° about the origin
   The transformation $(x, y) \to (-x, -y)$ represents a 180° rotation about the origin. This reverses the signs of both the x- and y-coordinates of $C$. Again, the exact coordinates of $C$ are needed to compute $C'$.

Work:

Translation:

Assume the coordinates of $B$ are $(b_x, b_y)$. After translation: $B' = (b_x - 4, b_y - 2)$

Rotation:

Assume the coordinates of $C$ are $(c_x, c_y)$. After a 180° rotation: $C' = (-c_x, -c_y)$

The results for $B'$ and $C'$ depend on the original points $B$ and $C$.

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If specific coordinates for $B$ and $C$ are provided, I can calculate the exact transformed coordinates. Let me know if you have those, or if you need further clarification!

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5 Related Questions:

1. What are the properties of a 180° rotation, and how does it compare to other rotations?
2. How does translation affect the slope of a line connecting two points?
3. Can translations and rotations be combined into a single transformation matrix?
4. How does the distance between two points change under translation and rotation?
5. How would the transformations change if the rotation was around a point other than the origin?

Tip: When solving problems involving transformations, always apply operations step by step to avoid mistakes.