Rotating Objects 180 Degrees Around The Origin

Review a quick way to rotate an object 180 degrees around the coordinate plane.


To rotate a triangle \( \text{ABC} \) by 180 degrees around the origin, you need to perform the following steps:

1. Identify the Original Coordinates
   \( A(-4,1) \)
   \( B(-4,5) \)
   \( C(-2,1) \)

2. Find the New Coordinates after Rotation
   To rotate a point \((x, y)\) by 180 degrees around the origin, you simply negate both \(x\) and \(y\) to find the new coordinates: \( (-x, -y) \).

3. Apply the Rotation to Each Vertex
   For point \( A(-4,1) \):
     \( A'(-(-4), -1) = (4, -1) \)
   For point \( B(-4,5) \):
     \( B'(-(-4), -5) = (4, -5) \)
   For point \( C(-2,1) \):
     \( C'(-(-2), -1) = (2, -1) \)

4. Plot the New Triangle
   Plot the points \( A'(4, -1) \), \( B'(4, -5) \), and \( C'(2, -1) \) on the coordinate plane to form the rotated triangle.

5. Connect the Points
   Connect the points \( A' \), \( B' \), and \( C' \) to form the rotated triangle \( A'B'C' \).

6. Verify the Rotation
   Check if the new triangle \( A'B'C' \) is rotated 180 degrees around the origin from the original triangle \( ABC \).

In summary, to rotate triangle \( ABC \) 180 degrees around the origin:
Point \( A(-4,1) \) becomes \( A'(4, -1) \).
Point \( B(-4,5) \) becomes \( B'(4, -5) \).
Point \( C(-2,1) \) becomes \( C'(2, -1) \).

 

 

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