Question

angle rst is rotated 180° counterclockwise about the origin. the result is △rst, 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 r(1, -5) → r(□,□) s(2, 1) → s(□,□) t(4, -7) → t(□,□) (b) choose the general rule below that describes the rotation mapping △rst to △rst. (x, y) → (-x, y) (x, y) → (y, x) (x, y) → (y, -x) (x, y) → (-y, x) (x, y) → (-y, -x) (x, y) → (x, -y) (x, y) → (-x, -y)

Explanation:

Step1: Recall the rule for \(180^{\circ}\) rotation

When a point \((x,y)\) is rotated \(180^{\circ}\) counter - clockwise about the origin, the rule is \((x,y)\to(-x,-y)\).

Step2: Apply the rule to point \(R(1, - 5)\)

For \(x = 1\) and \(y=-5\), using the rule \((x,y)\to(-x,-y)\), we get \(R'( - 1,5)\).

Step3: Apply the rule to point \(S(2,1)\)

For \(x = 2\) and \(y = 1\), using the rule \((x,y)\to(-x,-y)\), we get \(S'(-2,-1)\).

Step4: Apply the rule to point \(T(4,-7)\)

For \(x = 4\) and \(y=-7\), using the rule \((x,y)\to(-x,-y)\), we get \(T'(-4,7)\).

Answer:

(a) \(R(1,-5)\to R'(-1,5)\), \(S(2,1)\to S'(-2,-1)\), \(T(4,-7)\to T'(-4,7)\)
(b) The general rule is \((x,y)\to(-x,-y)\)