Question

triangle def is rotated 180° counterclockwise about the origin. the result is △def, 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 d(1, -5) → d() e(2, 1) → e() f(4, -7) → f() (b) choose the general rule below that describes the rotation mapping △def to △def (x, y) → (-y, x) (x, y) → (x, -y) (x, y) → (-x, -y) (x, y) → (-x, y) (x, y) → (y, x) (x, y) → (y, -x) (x, y) → (-y, -x)

Explanation:

Step1: Apply the rule for 180° rotation

The rule for a 180° counter - clockwise rotation about the origin is \((x,y)\to(-x,-y)\).
For point \(D(1,-5)\):
Substitute \(x = 1\) and \(y=-5\) into the rule \((x,y)\to(-x,-y)\).
We get \((-1,5)\).

Step2: Apply the rule for point \(E(2,1)\)

Substitute \(x = 2\) and \(y = 1\) into the rule \((x,y)\to(-x,-y)\).
We get \((-2,-1)\).

Step3: Apply the rule for point \(F(4,-7)\)

Substitute \(x = 4\) and \(y=-7\) into the rule \((x,y)\to(-x,-y)\).
We get \((-4,7)\).

Step4: Identify the general rule

Since the rule is \((x,y)\to(-x,-y)\), we check the options.

Answer:

(a) \(D(1,-5)\to D'(-1,5)\), \(E(2,1)\to E'(-2,-1)\), \(F(4,-7)\to F'(-4,7)\)
(b) \((x,y)\to(-x,-y)\)