Geometric Transformation Math Example 4

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Example 4

hard
Show that a 180° rotation about the origin is equivalent to the transformation (x,y)→(−x,−y)(x,y) \to (-x,-y). Verify with the point (3,4)(3, 4).

Solution

  1. 1
    Step 1: Use the rotation formula: (x′,y′)=(xcos⁡θ−ysin⁡θ, xsin⁡θ+ycos⁡θ)(x', y') = (x\cos\theta - y\sin\theta,\ x\sin\theta + y\cos\theta).
  2. 2
    Step 2: At θ=180°\theta=180°: cos⁡180°=−1\cos 180° = -1, sin⁡180°=0\sin 180° = 0.
  3. 3
    Step 3: (x′,y′)=(x(−1)−y(0), x(0)+y(−1))=(−x,−y)(x', y') = (x(-1) - y(0),\ x(0) + y(-1)) = (-x, -y).
  4. 4
    Step 4: Verify: (3,4)→(−3,−4)(3,4) \to (-3, -4).

Answer

(3,4)→(−3,−4)(3,4) \to (-3,-4). A 180° rotation equals (x,y)→(−x,−y)(x,y)\to(-x,-y).
The rotation matrix for angle θ\theta applied at 180°180° becomes (−100−1)\begin{pmatrix}-1 & 0 \\ 0 & -1\end{pmatrix}, which negates both coordinates. This transformation is also called a point reflection through the origin.

About Geometric Transformation

A function that maps every point of a geometric figure to a new position, changing its location, orientation, or size.

Learn more about Geometric Transformation →

More Geometric Transformation Examples

Example 1 easy

Name and describe the four basic geometric transformations.

Example 2 medium

Point [formula] is reflected over the y-axis, then translated by [formula]. Find the final image.

Example 3 easy

Which geometric transformations are isometries (preserve distances)?

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