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) → (\underline{\quad}, \underline{\quad}) rotation: (x, y) → (\underline{\quad}, \underline{\quad})

Explanation:

Step1: Find the reflection rule

For a reflection across the \(x -\)axis, the \(x -\)coordinate remains the same, and the \(y -\)coordinate changes its sign.
If we have a point \((x,y)\), after reflection across the \(x -\)axis, the image is \((x,-y)\).

Step2: Find the rotation rule

For a rotation of \(180^{\circ}\) about the origin, if we have a point \((x,y)\), the rule is \((x,y)\to(-x,-y)\). We can verify this using the general formula for a rotation of \(180^{\circ}\) about the origin. The rotation matrix \(

$$\begin{bmatrix}-1&0\\0& - 1\end{bmatrix}$$

\) when multiplied by the column vector \(

$$\begin{bmatrix}x\\y\end{bmatrix}$$

\) gives \(

$$\begin{bmatrix}-x\\-y\end{bmatrix}$$

\)

Answer:

Reflection: \((x,y)\mapsto(x, - y)\)
Rotation: \((x,y)\mapsto(-x,-y)\)