Question

(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) \to x(square,square) ) ( y(-6,7) \to y(square,square) ) ( z(-2,-3) \to z(square,square) ) (b) choose the general rule below that describes the rotation mapping ( \triangle xyz ) to ( \triangle xyz ). ( (x,y)\to(-x,-y) ) ( (x,y)\to(-y,x) ) ( (x,y)\to(y,-x) ) ( (x,y)\to(x,-y) ) ( (x,y)\to(-x,y) ) ( (x,y)\to(y,x) ) ( (x,y)\to(-y,-x) )

Explanation:

Step1: Determine coordinates from graph

From the graph, for point \(X(-8,1)\), the final coordinate \(X'\) is \((-1, - 8)\).
For point \(Y(-6,7)\), the final coordinate \(Y'\) is \((-7,-6)\).
For point \(Z(-2,-3)\), the final coordinate \(Z'\) is \((3,-2)\).

Step2: Check rotation rule

Let's check the rule \((x,y)\to(-y,x)\)
For \(X(-8,1)\): \(x=-8,y = 1\), then \(-y=-1,x=-8\) (not correct)
For the rule \((x,y)\to(-y,-x)\)
For \(X(-8,1)\): \(x=-8,y = 1\), \(-y=-1,-x = 8\) (not correct)
For the rule \((x,y)\to(y,-x)\)
For \(X(-8,1)\): \(x=-8,y = 1\), \(y = 1,-x=8\) (not correct)
For the rule \((x,y)\to(-x,-y)\)
For \(X(-8,1)\): \(-x = 8,-y=-1\) (not correct)
For the rule \((x,y)\to(x,-y)\)
For \(X(-8,1)\): \(x=-8,-y=-1\) (not correct)
For the rule \((x,y)\to(-x,y)\)
For \(X(-8,1)\): \(-x = 8,y = 1\) (not correct)
For the rule \((x,y)\to(y,x)\)
For \(X(-8,1)\): \(y = 1,x=-8\) (not correct)
Let's check the rule \((x,y)\to(-y,-x)\)
For \(X(-8,1)\): \(x=-8,y = 1\), \(-y=-1,-x = 8\) (not correct)
Let's use the coordinate - based approach.
If we assume a rotation of \(270^{\circ}\) counter - clockwise (or \(90^{\circ}\) clockwise) about the origin. The rule for a \(270^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(y,-x)\)
For \(X(-8,1)\): \(x=-8,y = 1\), \(y = 1,-x = 8\) (not correct)
The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\)
For \(X(-8,1)\): \(x=-8,y = 1\), \(-y=-1,x=-8\) (not correct)
Let's check by substitution:
If \(X(-8,1)\) and \(X'(-1,-8)\), \(Y(-6,7)\) and \(Y'(-7,-6)\), \(Z(-2,-3)\) and \(Z'(3,-2)\)
We can see that if \((x,y)\) is the original point and \((x',y')\) is the rotated point, \(x'=-y\) and \(y'=-x\)

Answer:

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