Question

triangle xyz has vertices x(1, 3), y(0, 0), and z(-1, 2). the image of triangle xyz after a rotation has vertices x(-3, 1), y(0, 0), and z(-2, -1). which rule describes the transformation?
ro, 90°
ro, 180°
ro, 270°
ro, 360°

Explanation:

Step1: Recall rotation rules

• For a rotation of \(90^{\circ}\) counter - clockwise about the origin \((x,y)\to(-y,x)\)
• For a rotation of \(180^{\circ}\) about the origin \((x,y)\to(-x,-y)\)
• For a rotation of \(270^{\circ}\) counter - clockwise about the origin \((x,y)\to(y, - x)\)
• For a rotation of \(360^{\circ}\) about the origin \((x,y)\to(x,y)\)

Step2: Check the transformation of a point

Take point \(X(1,3)\).
If we use the rule for a \(90^{\circ}\) counter - clockwise rotation \(R_{0,90^{\circ}}\):
Substitute \(x = 1\) and \(y = 3\) into \((x,y)\to(-y,x)\).
We get \((1,3)\to(-3,1)\)

Take point \(Z(-1,2)\).
Substitute \(x=-1\) and \(y = 2\) into \((x,y)\to(-y,x)\)
We get \((-1,2)\to(-2,-1)\)

Point \(Y(0,0)\):
Substitute \(x = 0\) and \(y = 0\) into \((x,y)\to(-y,x)\), we get \((0,0)\to(0,0)\)

Answer:

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