Question

question 5
what is the rule for a 180° rotation about the origin?
(x,y)→(y,−x)
(x,y)→(−y,x)
(x,y)→(x,−y)
(x,y)→(−x,−y)

Explanation:

When rotating a point \((x,y)\) \(180^{\circ}\) about the origin, both the \(x\) - coordinate and the \(y\) - coordinate change their signs.
For example, if we have a point \((2,3)\), after a \(180^{\circ}\) rotation about the origin, it becomes \((- 2,-3)\).
If we consider the general form of a point \((x,y)\) in the coordinate plane, a \(180^{\circ}\) rotation about the origin is equivalent to a reflection through the origin. Mathematically, this is represented as \((x,y)\to(-x,-y)\).
The rule \((x,y)\to(y, - x)\) is for a \(270^{\circ}\) counter - clockwise (or \(90^{\circ}\) clockwise) rotation about the origin.
The rule \((x,y)\to(-y,x)\) is for a \(90^{\circ}\) counter - clockwise (or \(270^{\circ}\) clockwise) rotation about the origin.
The rule \((x,y)\to(x,-y)\) is for a reflection over the \(x\) - axis.

Answer:

\((x,y)\to(-x,-y)\)