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 \(90^\circ\) counterclockwise rotation about the origin, the rule is \((x,y)\to(-y,x)\). For a \(270^\circ\) counterclockwise (or \(90^\circ\) clockwise) 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 Option A (270° CCW)

For point \(K(2,-1)\): Apply \((x,y)\to(y,-x)\). So \(y = -1\), \(-x=-2\), gives \((-1,-2)\), which matches \(K'(-1,-2)\).
For point \(L(4,1)\): Apply \((x,y)\to(y,-x)\). \(y = 1\), \(-x=-4\)? Wait, no, wait: Wait, \(270^\circ\) CCW is \((x,y)\to(y, -x)\)? Wait, no, correction: \(90^\circ\) CCW: \((x,y)\to(-y,x)\); \(180^\circ\) CCW: \((x,y)\to(-x,-y)\); \(270^\circ\) CCW: \((x,y)\to(y, -x)\). Wait, let's recalculate \(L(4,1)\) for \(270^\circ\) CCW: \(y = 1\), \(-x=-4\)? No, that's not matching \(L'(1,-4)\). Wait, wait, maybe I mixed up. Wait, \(270^\circ\) counterclockwise is equivalent to \(90^\circ\) clockwise, and the rule for \(90^\circ\) clockwise is \((x,y)\to(y, -x)\)? Wait, no, let's check standard rotation matrices. The rotation matrix for \(270^\circ\) counterclockwise is \(

$$\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$$

\), so applying to \((x,y)\) gives \((y, -x)\). Wait, for \(K(2,-1)\): \(x=2\), \(y=-1\), so \((y, -x)=(-1, -2)\), which is \(K'\). For \(L(4,1)\): \(x=4\), \(y=1\), so \((y, -x)=(1, -4)\), which is \(L'\)! Oh, I made a mistake earlier. So \(270^\circ\) CCW (or \(90^\circ\) CW) gives \((y, -x)\), which matches both \(K\to K'\) and \(L\to L'\).

Step3: Test Other Options

• Option B (90° CCW): Rule \((x,y)\to(-y,x)\). For \(K(2,-1)\): \(-y = 1\), \(x=2\), gives \((1,2)\), not \(K'(-1,-2)\). Eliminate B.
• Option C (Reflection over \(y\)-axis): \((x,y)\to(-x,y)\). For \(K(2,-1)\): \((-2,-1)\), not \(K'(-1,-2)\). Eliminate C.
• Option D (Reflection over \(x\)-axis): \((x,y)\to(x,-y)\). For \(K(2,-1)\): \((2,1)\), not \(K'(-1,-2)\). Eliminate D.

Answer:

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