Question

angle xyz is rotated 270° counterclockwise about the origin.\
e result is $\triangle xyz$, 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 $\to$ final coordinates\
\
$x(-5, -4) \to x(\underline{-4}, \underline{6})$\
\
$y(2, -2) \to y(\underline{-2}, \underline{-2})$\
\
$z(-3, -6) \to z(\underline{-6}, \underline{3})$\
\
(b) choose the general rule below that describes the rotation\
mapping $\triangle xyz$ to $\triangle xyz$.\
\
$\circ (x,y) \to (y, -x)$\
$\circ (x,y) \to (-y, -x)$\
$\circ (x,y) \to (-y, x)$\
$\circ (x,y) \to (-x, -y)$\
$\circ (x,y) \to (-x, y)$\
$\circ (x,y) \to (x, -y)$\
$\circ (x,y) \to (y, x)$

Explanation:

Step1: Verify 270° CCW rotation rule

A 270° counterclockwise rotation about the origin follows the rule: $(x, y) \to (y, -x)$.

Step2: Check point X transformation

Substitute $X(-5, -4)$ into the rule:
$(-5, -4) \to (-4, -(-5)) = (-4, 5)$

Step3: Check point Y transformation

Substitute $Y(2, -2)$ into the rule:
$(2, -2) \to (-2, -(2)) = (-2, -2)$

Step4: Check point Z transformation

Substitute $Z(-3, -6)$ into the rule:
$(-3, -6) \to (-6, -(-3)) = (-6, 3)$

Step5: Confirm correct general rule

The rule $(x, y) \to (y, -x)$ matches all coordinate transformations.

Answer:

(a)

$X(-5, -4) \to X'(-4, 5)$
$Y(2, -2) \to Y'(-2, -2)$
$Z(-3, -6) \to Z'(-6, 3)$

(b)

$\boldsymbol{(x, y) \to (y, -x)}$