Question

triangle hij is the image of triangle hij under a reflection across the x-axis followed by a rotation about the origin.
write the rules for the reflection and rotation.
reflection: $(x, y) \
ightarrow (\square, \square)$
rotation: $(x, y) \
ightarrow (\square, \square)$

Explanation:

Step1: Identify reflection rule

Reflection over x-axis flips y-sign.
Rule: $(x, y) \to (x, -y)$

Step2: Pick a point to test rotation

Take original point $H(-9, 2)$. Apply reflection: $(-9, -2)$. The final image $H'$ is $(0, -9)$.

Step3: Find rotation mapping

Map $(-9, -2)$ to $(0, -9)$: this is a 90° clockwise rotation about origin, which follows $(x, y) \to (y, -x)$. Verify with $I(-6, 6)$: reflection gives $(-6, -6)$, rotation gives $(-6, 6)$ which matches $I'(6, -6)$.

Answer:

Reflection: $(x, y) \to (x, -y)$
Rotation: $(x, y) \to (y, -x)$