Question

angle xyz is rotated 270° counterclockwise about the origin. the result is △xyz, as shown below. (a) the arrows below show that the coordinates on the left are mapped to the coordinates on the right. fill in the blanks to give the coordinates after the rotation. original coordinates → final coordinates x(-8,1)→x(□,□) y(-6,7)→y(□,□) z(-2,-3)→z(□,□) (b) choose the general rule below that describes the rotation mapping △xyz to △xyz. (x,y)→(y,-x) (x,y)→(-y,x) (x,y)→(-y,-x) (x,y)→(-x,-y) (x,y)→(-x,y) (x,y)→(y,x) (x,y)→(x,-y)

Explanation:

Step1: Apply rotation rule

The rule for a \(270^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(y, - x)\).

Step2: Calculate \(X'\)

For \(X(-8,1)\), using the rule \((x,y)\to(y, - x)\), we substitute \(x=-8\) and \(y = 1\). So \(X'=(1,8)\).

Step3: Calculate \(Y'\)

For \(Y(-6,7)\), substitute \(x=-6\) and \(y = 7\) into the rule \((x,y)\to(y, - x)\). Then \(Y'=(7,6)\).

Step4: Calculate \(Z'\)

For \(Z(-2,-3)\), substitute \(x=-2\) and \(y=-3\) into the rule \((x,y)\to(y, - x)\). So \(Z'=(-3,2)\).

Answer:

(a) \(X'=(1,8)\), \(Y'=(7,6)\), \(Z'=(-3,2)\)
(b) \((x,y)\to(y, - x)\)