Question 9
Given $\overline{B C}$, describe the error made when rotating $B(3,-1)$ counterclockwise $270^{\circ}$ about $C(-2,-5)$.
- Step 1: $(3,-1) \rightarrow(1,-6)$
- Step 2: $(1,-6) \rightarrow(-6,-1)$
- Step 3: $(-6,-1) \rightarrow(-4,4)$

The image is located at $B^{\prime}(-4,4)$.

Step 1 ：Given points B(3,-1) and C(-2,-5), and a rotation of 270 degrees counterclockwise about point C.

Step 2 ：The general formula to rotate a point (x, y) about another point (p, q) by an angle θ counterclockwise is given by: \(x' = p + (x - p) * cos(θ) - (y - q) * sin(θ)\) and \(y' = q + (x - p) * sin(θ) + (y - q) * cos(θ)\).

Step 3 ：Using this formula, the correct coordinates of the point B after a 270 degrees counterclockwise rotation about point C are calculated to be approximately (2, -10).

Step 4 ：However, according to the steps provided in the question, the image is located at B'(-4,4).

Step 5 ：Therefore, the error made when rotating B(3,-1) counterclockwise 270 degrees about C(-2,-5) is that the final coordinates of the point B' are incorrect.

Step 6 ：Final Answer: The error made when rotating B(3,-1) counterclockwise 270 degrees about C(-2,-5) is that the final coordinates of the point B' are incorrect. The correct coordinates should be approximately \(\boxed{(2, -10)}\), but the image is located at \(\boxed{B'(-4,4)}\).