Question

1. given the points below, which of the following transformations maps kl to kl?

k(2, -1) k(-1, -2)
l(4, 1) l(1, -4)
a. 270° counterclockwise rotation about the origin
b. 90° counterclockwise rotation about the origin
c. reflection across the y - axis
d. reflection across the x - axis

Explanation:

Step1: Recall Rotation Rules

For a \(270^\circ\) counterclockwise (or \(90^\circ\) clockwise) rotation about the origin, the rule is \((x,y)\to(y, -x)\). For a \(90^\circ\) counterclockwise rotation, the rule is \((x,y)\to(-y,x)\). For reflection over \(y\)-axis: \((x,y)\to(-x,y)\), over \(x\)-axis: \((x,y)\to(x,-y)\).

Step2: Test \(270^\circ\) Counterclockwise on \(K(2,-1)\)

Using \((x,y)\to(y, -x)\): \(x = 2\), \(y=-1\). So \(y=-1\), \(-x=-2\). Thus, \((-1,-2)\), which matches \(K'(-1,-2)\).

Step3: Test \(270^\circ\) Counterclockwise on \(L(4,1)\)

Using \((x,y)\to(y, -x)\): \(x = 4\), \(y = 1\). So \(y = 1\), \(-x=-4\). Thus, \((1,-4)\), which matches \(L'(1,-4)\).

Step4: Eliminate Other Options

• For \(90^\circ\) counterclockwise on \(K(2,-1)\): \((-(-1),2)=(1,2)

eq K'(-1,-2)\).

• For reflection over \(y\)-axis on \(K(2,-1)\): \((-2,-1)

eq K'(-1,-2)\).

• For reflection over \(x\)-axis on \(K(2,-1)\): \((2,1)

eq K'(-1,-2)\).

Answer:

A. \(270^\circ\) counterclockwise rotation about the origin