Complete a glide reflection to the point H(-2, -3) with the translation (x, y) → (x – 2, y) and a reflection over y = -4.

Question

Find the point H”.

Answer 1

To complete a glide reflection for the point $H(-2,-3)$ using the translation $(x,y)\to (x-2,y)$ and a reflection over the line $y=-4$, we'll follow these steps:

1. Apply the translation: The translation $(x,y)\to (x-2,y)$ transforms point H(-2, -3) as follows:

   $$H^{\prime }=(-2-2,-3)=(-4,-3)$$

2. Reflect over the line $y=-4$: To reflect the point $H^{\prime }(-4,-3)$ over the line $y=-4$, we need to find the distance from $H^{\prime }$ to the line $y=-4$.

   • The y-coordinate of $H^{\prime }$ is $-3$.
   • The line $y=-4$ is below $H^{\prime }$ (since $-3>-4$).
   • The distance from $H^{\prime }$ to the line $y=-4$ is:

   $$\text{Distance}=-3-(-4)=-3+4=1$$

   This means the reflected point $H^{″}$ will be 1 unit below the line $y=-4$:

   $$H^{″}=(-4,-4-1)=(-4,-5)$$

Hence, the point $H^{″}$ after performing the glide reflection is:

$$\boxed{{\displaystyle (-4,-5)}}$$