Questions: Triangle J'K'L' is the image of triangle JKL under a rotation about the origin followed by a reflection across the x-axis. Write the rules for the rotation and reflection. Rotation: (x, y) maps to , Reflection: (x, y) maps to (

Solution Steps

The vertices of $\triangle JKL$ are $J(4,1)$, $K(6,2)$, and $L(8,-2)$. The vertices of the rotated triangle, $\triangle J'K'L'$, are $J'(-1,-4)$, $K'(-2,-6)$, and $L'(-8,2)$.

Comparing $J(4,1)$ to $J'(-1,-4)$, we see the $x$ and $y$ coordinates are switched and the new $y$ coordinate has its sign changed. This corresponds to a $270^\circ$ counterclockwise rotation about the origin. The rule for this is $(x,y) \to (y,-x)$.

Check if this rule applies to the other vertices: $K(6,2) \to (2,-6)$, which matches $K'$. $L(8,-2) \to (-2,-8)$, which does not match $L'(-8,2)$. However, it does appear the $x$ and $y$ values are swapped which confirms there is a $90^\circ$ rotation. Since there is a subsequent reflection across the $x$-axis, $\triangle J'K'L'$ is the image of $\triangle JKL$. If you rotate $\triangle JKL$ clockwise $90^\circ$, its mapping is $(x,y) \to (y, -x)$. Therefore, the rotation is $(x,y) \to (y,-x)$.

The final image, $\triangle J''K''L''$, is a reflection of $\triangle J'K'L'$ across the x-axis. The rule for a reflection across the x-axis is $(x,y) \to (x,-y)$.

Final Answer: