A point P(5, 1) is rotated by 90° about a pivot point (2, 2). What is the coordinate of new transformed point P′ ?
(3, 5)
Rotation about a pivot point \((x_r, y_r)\) is a composite transformation: translate the pivot to the origin, rotate, then translate back. A positive angle means anticlockwise rotation: \(x' = x_r + (x - x_r)\cos\theta - (y - y_r)\sin\theta\), \(y' = y_r + (x - x_r)\sin\theta + (y - y_r)\cos\theta\).
Step 1 — Translate: \((5 - 2,\ 1 - 2) = (3, -1)\).
Step 2 — Rotate by 90° with \(\cos 90^\circ = 0,\ \sin 90^\circ = 1\): \((3 \cdot 0 - (-1) \cdot 1,\ 3 \cdot 1 + (-1) \cdot 0) = (1, 3)\).
Step 3 — Translate back: \((1 + 2,\ 3 + 2) = (3, 5)\).
Check: the distance from the pivot is \(\sqrt{3^2 + 1^2} = \sqrt{10}\) before and after. (2, 4) is not at distance \(\sqrt{10}\) from (2, 2); (5, 3) and (1, 5) are at that distance but correspond to rotations by about 37° and 127° rather than 90°.
(3, 5) option is correct.
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