Question

after a rotation, ( a(-3,4) ) maps to ( a(4,3) ), ( b(4,-5) ) maps to ( b(-5,-4) ), and ( c(1,6) ) maps to ( c(6,-1) ). which rule describes the rotation?
( r_{0,90^{circ}} )
( r_{0,180^{circ}} )
( r_{0,270^{circ}} )
( r_{0,360^{circ}} )

Explanation:

Step1: Recall rotation rules

• For a \(90^{\circ}\) rotation about the origin \((x,y)\to(-y,x)\)
• For a \(180^{\circ}\) rotation about the origin \((x,y)\to(-x,-y)\)
• For a \(270^{\circ}\) rotation about the origin \((x,y)\to(y, -x)\)
• For a \(360^{\circ}\) rotation about the origin \((x,y)\to(x,y)\)

Step2: Check each point

• For \(A(-3,4)\):
• If we use the rule \((x,y)\to(y, -x)\) ( \(270^{\circ}\) rotation), when \(x = - 3\) and \(y = 4\), we get \((4,3)\)
• For \(B(4,-5)\):
• When \(x = 4\) and \(y=-5\), using the rule \((x,y)\to(y, -x)\), we get \((-5,-4)\)
• For \(C(1,6)\):
• When \(x = 1\) and \(y = 6\), using the rule \((x,y)\to(y, -x)\), we get \((6,-1)\)

Answer:

\(R_{0,270^{\circ}}\)