Question

true or false: rotating ( m(-5,1) 180^{0} ) and then translating along ( langle-3,4
angle ) will give the same result as translating along ( langle-3,4
angle ) and then rotating ( 180^{0} ).
o true
o false

Explanation:

Step1: Rotate then translate

• Rotate \(M(-5,1)\) \(180^{\circ}\): The rule for a \(180^{\circ}\) rotation is \((x,y)\to(-x,-y)\). So \(M(-5,1)\to M'(5, - 1)\).
• Translate \(M'(5,-1)\) along \(\langle-3,4

angle\): The rule for translation \(\langle a,b
angle\) is \((x,y)\to(x + a,y + b)\). So \(M'(5,-1)\to M''(5-3,-1 + 4)=(2,3)\).

Step2: Translate then rotate

• Translate \(M(-5,1)\) along \(\langle-3,4

angle\): Using \((x,y)\to(x + a,y + b)\), we get \(M(-5,1)\to M_1(-5-3,1 + 4)=(-8,5)\).

• Rotate \(M_1(-8,5)\) \(180^{\circ}\): Using \((x,y)\to(-x,-y)\), we get \(M_1(-8,5)\to M_2(8,-5)\).

Answer:

False