Question

writing a rule for a rotation
triangle abc was rotated about the origin. 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 \(R_{0,90^{\circ}}\): \((x,y)\to(-y,x)\)
• For \(R_{0,180^{\circ}}\): \((x,y)\to(-x,-y)\)
• For \(R_{0,270^{\circ}}\): \((x,y)\to(y, -x)\)
• For \(R_{0,360^{\circ}}\): \((x,y)\to(x,y)\)

Step2: Identify coordinates

• Let \(A=(1,4)\), \(B=(3,4)\), \(C=(3,2)\)
• After rotation \(A'=(-4,1)\), \(B'=(-4,3)\), \(C'=(-2,3)\)

Step3: Apply rotation rules

• Take point \(A=(1,4)\)
• If we apply \(R_{0,270^{\circ}}\): \((x = 1,y = 4)\to(y,-x)=(4,-1)\) (incorrect)
• If we apply \(R_{0,90^{\circ}}\): \((x = 1,y = 4)\to(-y,x)=(-4,1)\) (correct for \(A\))
• Check another point \(B=(3,4)\), \(R_{0,90^{\circ}}\): \((x = 3,y = 4)\to(-y,x)=(-4,3)\) (correct for \(B'\))
• Check \(C=(3,2)\), \(R_{0,90^{\circ}}\): \((x = 3,y = 2)\to(-y,x)=(-2,3)\) (correct for \(C'\))

Answer:

\(R_{0,90^{\circ}}\)