Problem

5. Reflect $\triangle A B C$ over the $y$-axis, translate by $\langle 2,-1)$, and rotate the result $180^{\circ}$ about the origin. Plot $\triangle A^{\prime \prime \prime} B^{\prime \prime \prime} C^{\prime \prime \prime}$ on the grid below. ( 2 points)
Transformation rule: A=2,-1 B=2,-4 C=4,-2

Answer

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Answer

\(\boxed{\text{The final coordinates of the transformed triangle are A' = (0, 2), B' = (0, 5), and C' = (2, 3).}}\)

Steps

Step 1 :Reflect the points A = [2, -1], B = [2, -4], C = [4, -2] over the y-axis. This changes the sign of the x-coordinate of each point, resulting in A1 = [-2, -1], B1 = [-2, -4], C1 = [-4, -2].

Step 2 :Translate the points by the vector <2,-1>. This means we add 2 to the x-coordinate and subtract 1 from the y-coordinate of each point, resulting in A2 = [0, -2], B2 = [0, -5], C2 = [-2, -3].

Step 3 :Rotate the points 180 degrees about the origin. This changes the sign of both the x and y coordinates of each point, resulting in A3 = [0, 2], B3 = [0, 5], C3 = [2, 3].

Step 4 :\(\boxed{\text{The final coordinates of the transformed triangle are A' = (0, 2), B' = (0, 5), and C' = (2, 3).}}\)

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