Question

triangle def is rotated 90° counterclockwise about the origin. the result is △d′e′f′, 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(-5, 3) → d(-3, -5) e(-1, 1) → e(-1, -1) f(-8, -4) → f(4, -8) (b) choose the general rule below that describes the rotation mapping △def to △d′e′f′. ○ (x, y) → (x, -y) ○ (x, y) → (y, x) ○ (x, y) → (-x, -y) ○ (x, y) → (y, -x) ○ (x, y) → (-y, x) ○ (x, y) → (-y, -x) ○ (x, y) → (-x, y)

Explanation:

Step1: Verify 90° CCW rotation rule

For a 90° counterclockwise rotation about the origin, the rule is $(x,y) \to (-y,x)$.

Step2: Check point D substitution

Substitute $D(-5,3)$: $(-3, -(-5)) = (-3,5)$? No, wait, correct substitution: $(-y,x) = (-(3), -5) = (-3,-5)$, which matches $D'(-3,-5)$.

Step3: Check point E substitution

Substitute $E(-1,1)$: $(-1, -(-1)) = (-1,1)$? No, correct: $(-y,x) = (-(1), -1) = (-1,-1)$, which matches $E'(-1,-1)$.

Step4: Check point F substitution

Substitute $F(-8,-4)$: $(-(-4), -8) = (4,-8)$, which matches $F'(4,-8)$.

Step5: Confirm general rule

The rule $(x,y) \to (-y,x)$ works for all points.

Answer:

(a)
$D(-5, 3) \to D'(-3, -5)$
$E(-1, 1) \to E'(-1, -1)$
$F(-8, -4) \to F'(4, -8)$

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