How do you rotate 180 degrees not around the origin? - Answers

Graphically: Measure the distance from each point ot the centre of rotation and continue to the other side. This is easiest done by measuring the x and y distances separately; they swap sides of the point: left ←→ right, above ←→ below. eg: A triangle ABC {(1,1), (3,4), (2,1)} rotated 180° about point (2, 2): ! (1, 1): x distance is 2 - 1 = 1 to left of centre, so new x is to right at 2 + 1 = 3 y distance is 2 - 1 = 1 below centre, so new y is above at 2 + 1 = 3 → A' is (3, 3) B (3, 4) x distance is 3 - 2 = 1 to right of centre, so new x is to left at 2 - 1 = 1 y distance is 4 - 2 = 2 above centre, so new y is below at 2 - 2 = 0 → B' is (1, 0) C (2, 1) x distance is 2 - 2 = 0 on the centre, so new x is also on the centre at 2 + 0 = 2 y distance is 2 - 1 = 1 below centre, so new y is above at 2 + 1 = 3 → C' is (2, 3) Thus triangle ABC {(1,1), (3,4), (2,1)} goes to triangle A'B'C' {(3,3), (1,0), (2,3)} when rotated 180° about centre (2,2). Algebraically: Rotating 180° about point (x0, y0): point (x, y) → (2 x0 - x, 2 y0 - y) For triangle ABC {(1,1), (3,4), (2,1)} rotated 180° about point (2, 2): A': (2×2 - 1, 2×2 - 1) = (3, 3) B': (2×2 - 3, 2×2 - 4) = (1, 0) C': (2×2 - 2, 2×2 - 1) = (2, 3) ie ABC → A'B'C {(3,3), (1,0), (2,3)} [as before].