Question

2 find the following rotations of (-5, 6) a. $r_{90}$ b. $r_{-90}$ c. $r_{270}$ d. $r_{-270}$ e. $r_{180}$

Explanation:

Step1: Recall Rotation Rules

For a point \((x,y)\):

• \(R_{90^\circ}\) (counterclockwise 90°): \((x,y)\to(-y,x)\)
• \(R_{-90^\circ}\) (clockwise 90° or counterclockwise 270°): \((x,y)\to(y,-x)\)
• \(R_{180^\circ}\): \((x,y)\to(-x,-y)\)
• \(R_{270^\circ}\) (counterclockwise 270° or clockwise 90°): \((x,y)\to(y,-x)\) (same as \(R_{-90^\circ}\))
• \(R_{-270^\circ}\) (clockwise 270° or counterclockwise 90°): \((x,y)\to(-y,x)\) (same as \(R_{90^\circ}\))

Step2: Apply \(R_{90^\circ}\) to \((-5,6)\)

Using \(R_{90^\circ}\) rule \((x,y)\to(-y,x)\):
\(x = -5\), \(y = 6\)
New point: \((-6,-5)\)

Step3: Apply \(R_{-90^\circ}\) to \((-5,6)\)

Using \(R_{-90^\circ}\) rule \((x,y)\to(y,-x)\):
\(x = -5\), \(y = 6\)
New point: \((6,5)\)

Step4: Apply \(R_{270^\circ}\) to \((-5,6)\)

Using \(R_{270^\circ}\) rule (same as \(R_{-90^\circ}\)) \((x,y)\to(y,-x)\):
New point: \((6,5)\) (same as \(R_{-90^\circ}\))

Step5: Apply \(R_{-270^\circ}\) to \((-5,6)\)

Using \(R_{-270^\circ}\) rule (same as \(R_{90^\circ}\)) \((x,y)\to(-y,x)\):
New point: \((-6,-5)\) (same as \(R_{90^\circ}\))

Step6: Apply \(R_{180^\circ}\) to \((-5,6)\)

Using \(R_{180^\circ}\) rule \((x,y)\to(-x,-y)\):
\(x = -5\), \(y = 6\)
New point: \((5,-6)\)

Answer:

A. \(R_{90^\circ}\): \(\boldsymbol{(-6, -5)}\)
B. \(R_{-90^\circ}\): \(\boldsymbol{(6, 5)}\)
C. \(R_{270^\circ}\): \(\boldsymbol{(6, 5)}\)
D. \(R_{-270^\circ}\): \(\boldsymbol{(-6, -5)}\)
E. \(R_{180^\circ}\): \(\boldsymbol{(5, -6)}\)