Question

triangle def is rotated 270° 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(3, -7) → d( ) e(4, 1) → e( ) f(7, -2) → f( ) (b) choose the general rule below that describes the rotation mapping △def to △def. (x, y) → (x, -y) (x, y) → (-x, -y) (x, y) → (-y, x) (x, y) → (-x, y) (x, y) → (y, -x) (x, y) → (-y, -x) (x, y) → (y, x)

Explanation:

Step1: Recall rotation rule

The rule for a 270 - degree counter - clockwise rotation about the origin is $(x,y)\to(y, - x)$.

Step2: Apply rule to point D

For point $D(3,-7)$, using the rule $(x,y)\to(y, - x)$, we substitute $x = 3$ and $y=-7$. So $D'( - 7,-3)$.

Step3: Apply rule to point E

For point $E(4,1)$, substitute $x = 4$ and $y = 1$ into the rule $(x,y)\to(y, - x)$. So $E'(1,-4)$.

Step4: Apply rule to point F

For point $F(7,-2)$, substitute $x = 7$ and $y=-2$ into the rule $(x,y)\to(y, - x)$. So $F'( - 2,-7)$.

Step5: Identify general rule

The general rule for a 270 - degree counter - clockwise rotation about the origin is $(x,y)\to(y, - x)$.

Answer:

(a) $D(3,-7)\to D'(-7,-3)$
$E(4,1)\to E'(1,-4)$
$F(7,-2)\to F'(-2,-7)$
(b) $(x,y)\to(y, - x)$