Question

rule 1
old → new
(1,1) → (-1,1)
(5,3) → (-3,5)
(7,5) → (-5,7)
this is a 90 degree counterclockwise rotation around the origin.
can you write the rule?
the rule is (x,y) →...

Explanation:

Step1: Analyze given points

Take the point \((1,1)\) old, new is \((-1,1)\)? Wait, no, wait the new points: \((1,1)\) old, new is \((-1,1)\)? Wait no, looking at the table: \((1,1) \to (-1,1)\)? Wait no, wait the blue and red figures. Wait the rule for 90° counterclockwise rotation about origin is \((x,y) \to (-y,x)\). Let's check with the points. Take \((5,3)\): old \(x = 5\), \(y = 3\). Applying \((-y,x)\) gives \((-3,5)\), which matches the new point \((-3,5)\). Check \((7,5)\): \(x = 7\), \(y = 5\), so \((-5,7)\), which matches. Check \((1,1)\): \(x = 1\), \(y = 1\), so \((-1,1)\), which matches. So the rule is \((x,y) \to (-y,x)\).

Step2: Confirm with rotation formula

The standard formula for a 90 - degree counterclockwise rotation about the origin is \((x,y)\) maps to \((-y,x)\). All given point transformations satisfy this: for \((x,y)\), new \(x\) is \(-y\), new \(y\) is \(x\).

Answer:

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