Question

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

Explanation:

Step1: Find the result of rotating then translating

• Rotation formula: When a point \((x,y)\) is rotated \(180^{\circ}\) about the origin, the new coordinates \((x',y')\) are given by \((x',y')=(-x,-y)\).

For the point \(M(-5,1)\), after a \(180^{\circ}\) rotation, the point becomes \(M_1=(5, - 1)\).

• Translation formula: When a point \((x,y)\) is translated along the vector \(\langle a,b

angle\), the new coordinates \((x'',y'')=(x + a,y + b)\).
After translating \(M_1=(5,-1)\) along \(\langle-3,4
angle\), we have \(x''=5+( - 3)=2\) and \(y''=-1 + 4=3\). So the final point is \((2,3)\).

Step2: Find the result of translating then rotating

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

angle\). Using the translation formula \((x,y)\to(x + a,y + b)\), we get \(x'=-5+( - 3)=-8\) and \(y'=1 + 4=5\). The point after translation is \(M_2=(-8,5)\).

• Rotation next: Rotate \(M_2=(-8,5)\) \(180^{\circ}\) about the origin. Using the rotation formula \((x,y)\to(-x,-y)\), we get \(x''=-(-8)=8\) and \(y''=-5\). So the final point is \((8,-5)\).

Answer:

False