How do you rotate 180 degrees not around the origin?

Answer 1

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].