Apply coordinate rules for rotations

A rotation about the origin through 90°, 180°, or 270° is represented by changing an ordered pair according to its direction: $(x,y)$ maps to $(-y,x)$ , $(-x,-y)$ , or $(y,-x)$ for counterclockwise rotations, with clockwise rotations reversing the corresponding direction. The understanding includes applying these rules to every vertex of a figure, tracking coordinate signs, and recognizing that rotations preserve lengths and angle measures; rotations about other centers, arbitrary angles, and trigonometric rules are outside this scope.

Detailed Explanation: Apply coordinate rules for rotations

A rotation moves every point the same amount around the origin. For common rotations, use these coordinate rules:

$90^\circ$ counterclockwise: $(x,y)\rightarrow(-y,x)$
$180^\circ$ : $(x,y)\rightarrow(-x,-y)$
$270^\circ$ counterclockwise: $(x,y)\rightarrow(y,-x)$

For a clockwise rotation, use the rule for the opposite direction. For example, a $90^\circ$ clockwise rotation uses $(x,y)\rightarrow(y,-x)$ .

Example

Rotate triangle $ABC$ $90^\circ$ counterclockwise about the origin.

A(2,1),\qquad B(4,1),\qquad C(2,3)

For a $90^\circ$ counterclockwise rotation, replace $(x,y)$ with $(-y,x)$ .

Apply the rule to each vertex:

Point $A(2,1)$ :

A'(2,1)\rightarrow(-1,2)

Point $B(4,1)$ :

B'(4,1)\rightarrow(-1,4)

Point $C(2,3)$ :

C'(2,3)\rightarrow(-3,2)

Therefore, the rotated triangle has vertices

\boxed{A'(-1,2),\quad B'(-1,4),\quad C'(-3,2)}

Be careful to change the signs and switch the order of the coordinates exactly as the rule shows. A rotation changes the location of a figure but preserves its side lengths and angle measures.

Learn by doing: Apply coordinate rules for rotations

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Cartesian Grid - Rotation of Point (Grid to Coordinates) around Origin

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