Final answer:
The reflected and rotated coordinate A' of the original point A (8, 2) is found to be at (7, -6). This involves reflecting A over the line y = 4 and then rotating it 90° around the axis point (1, 2).
Step-by-step explanation:
To find the reflection of coordinate A at (8, 2) across the line y = 4, we first determine the distance of A above the line y = 4. Since A is 2 units above the x-axis and the line y = 4 is 4 units above the x-axis, A is 2 units (4 - 2) below y = 4. To reflect A across y = 4, we move A up by twice this distance (4 units) to get the reflected point A' at (8, 6).
Next, we perform a 90° rotation of A' around the axis point (1, 2). To do this, we effectively swap the x and y values of A' and then change the sign of the new y-value to achieve a clockwise rotation. The original A' (8, 6) when swapped is (6, 8) and after changing the sign of the new y-value, we obtain the final coordinates (6, -8). However, since we rotate around the point (1, 2), we need to adjust by this pivot point. After the pivot adjustment, we end up with the final coordinate A’ at (6 + 1, -8 + 2), which simplifies to A’ at (7, -6).