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 (\(R_{0,90^{\circ}}\)): \((x,y)\to(-y,x)\)
• For a \(180^{\circ}\) rotation (\(R_{0,180^{\circ}}\)): \((x,y)\to(-x,-y)\)
• For a \(270^{\circ}\) rotation (\(R_{0,270^{\circ}}\)): \((x,y)\to(y,-x)\)
• For a \(360^{\circ}\) rotation (\(R_{0,360^{\circ}}\)): \((x,y)\to(x,y)\)

Step2: Apply the rules to the points

• For point \(A(-3,4)\)
• If we use \(R_{0,270^{\circ}}\): \((x = - 3,y = 4)\to(y,-x)=(4,3)\) (matches \(A'\))
• For point \(B(4,-5)\)
• If we use \(R_{0,270^{\circ}}\): \((x = 4,y=-5)\to(y,-x)=(-5,-4)\) (matches \(B'\))
• For point \(C(1,6)\)
• If we use \(R_{0,270^{\circ}}\): \((x = 1,y = 6)\to(y,-x)=(6,-1)\) (matches \(C'\))

Since the transformation \((x,y)\to(y,-x)\) (which is the rule for \(R_{0,270^{\circ}}\)) maps \(A(-3,4)\) to \(A'(4,3)\), \(B(4,-5)\) to \(B'(-5,-4)\) and \(C(1,6)\) to \(C'(6,-1)\), the rotation is \(R_{0,270^{\circ}}\)

Answer:

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