Rotation of 4 Vertices around Any Point (A) - Free Printable

To solve these rotation problems, we need to understand how coordinates change when you rotate a shape around a specific point.

Let’s break down the rules for rotating 90° clockwise and 90° counterclockwise around a point (h, k):

For a 90° clockwise rotation about (h, k):
- New x = h + (original y - k)
- New y = k - (original x - h)

For a 90° counterclockwise rotation about (h, k):
- New x = h - (original y - k)
- New y = k + (original x - h)

For a 180° rotation about (h, k):
- New x = h - (original x - h) = 2h - original x
- New y = k - (original y - k) = 2k - original y

Now let’s apply this to each problem.

Problem 1: Rotate 90° clockwise about (3, 1)
The triangle has vertices at approximately (2, -2), (4, -3), and (1, -4).

Using the formula for 90° clockwise about (3,1):
New x = 3 + (y - 1)
New y = 1 - (x - 3)

For (2, -2):
New x = 3 + (-2 - 1) = 3 + (-3) = 0
New y = 1 - (2 - 3) = 1 - (-1) = 2
→ (0, 2)

For (4, -3):
New x = 3 + (-3 - 1) = 3 + (-4) = -1
New y = 1 - (4 - 3) = 1 - 1 = 0
→ (-1, 0)

For (1, -4):
New x = 3 + (-4 - 1) = 3 + (-5) = -2
New y = 1 - (1 - 3) = 1 - (-2) = 3
→ (-2, 3)

Problem 2: Rotate 90° clockwise about (3, -3)
The triangle has vertices at approximately (1, -4), (5, -3), and (3, -5).

Using the same formula:

For (1, -4):
New x = 3 + (-4 - (-3)) = 3 + (-1) = 2
New y = -3 - (1 - 3) = -3 - (-2) = -1
→ (2, -1)

For (5, -3):
New x = 3 + (-3 - (-3)) = 3 + 0 = 3
New y = -3 - (5 - 3) = -3 - 2 = -5
→ (3, -5)

For (3, -5):
New x = 3 + (-5 - (-3)) = 3 + (-2) = 1
New y = -3 - (3 - 3) = -3 - 0 = -3
→ (1, -3)

Problem 3: Rotate 90° counterclockwise about (1, 0)
The triangle has vertices at approximately (0, 2), (3, 1), and (-1, 3).

Using the counterclockwise formula:
New x = 1 - (y - 0) = 1 - y
New y = 0 + (x - 1) = x - 1

For (0, 2):
New x = 1 - 2 = -1
New y = 0 - 1 = -1
→ (-1, -1)

For (3, 1):
New x = 1 - 1 = 0
New y = 3 - 1 = 2
→ (0, 2)

For (-1, 3):
New x = 1 - 3 = -2
New y = -1 - 1 = -2
→ (-2, -2)

Problem 4: Rotate 180° about (2, -4)
The triangle has vertices at approximately (0, -5), (4, -6), and (1, -7).

Using the 180° formula:
New x = 2*2 - x = 4 - x
New y = 2*(-4) - y = -8 - y

For (0, -5):
New x = 4 - 0 = 4
New y = -8 - (-5) = -8 + 5 = -3
→ (4, -3)

For (4, -6):
New x = 4 - 4 = 0
New y = -8 - (-6) = -8 + 6 = -2
→ (0, -2)

For (1, -7):
New x = 4 - 1 = 3
New y = -8 - (-7) = -8 + 7 = -1
→ (3, -1)

Final Answer:
The rotated images should be drawn using these new coordinates:
1. For 90° clockwise about (3,1): (0,2), (-1,0), (-2,3)
2. For 90° clockwise about (3,-3): (2,-1), (3,-5), (1,-3)
3. For 90° counterclockwise about (1,0): (-1,-1), (0,2), (-2,-2)
4. For 180° about (2,-4): (4,-3), (0,-2), (3,-1)