To find the coordinates of the vertices $B$, $C$, $D$, and $E$ after a reflection over the line $y = 1$, we need to follow these steps:

1. Identify the coordinates of each vertex.

   • $B = (-8, -8)$
   • $C = (-7, -2)$
   • $D = (-4, 3)$
   • $E = (-5, -3)$

2. Reflect each point across the line $y = 1$.

   • When reflecting a point $(x, y)$ over a horizontal line $y = k$, the reflected point will be $(x, 2k - y)$.

Since $y = 1$, we can apply this formula to each vertex.

Calculations

• Point $B = (-8, -8)$: $y' = 2(1) - (-8) = 2 + 8 = 10$ Reflected point of $B$: $(-8, 10)$.

• Point $C = (-7, -2)$: $y' = 2(1) - (-2) = 2 + 2 = 4$ Reflected point of $C$: $(-7, 4)$.

• Point $D = (-4, 3)$: $y' = 2(1) - 3 = 2 - 3 = -1$ Reflected point of $D$: $(-4, -1)$.

• Point $E = (-5, -3)$: $y' = 2(1) - (-3) = 2 + 3 = 5$ Reflected point of $E$: $(-5, 5)$.

Reflected Coordinates

After reflecting over the line $y = 1$, the coordinates of the vertices are:

• $B' = (-8, 10)$
• $C' = (-7, 4)$
• $D' = (-4, -1)$
• $E' = (-5, 5)$

Would you like further details on how reflections work, or have any specific questions?

Here are some follow-up questions:

1. How would the coordinates change if the reflection line was $y = -1$ instead of $y = 1$?
2. What would be the new coordinates if the shape was rotated 90 degrees around the origin?
3. How can we generalize the reflection formula for any line $y = k$?
4. What if we wanted to reflect over a vertical line, say $x = -4$?
5. How would you determine the distance between the original points and their reflected images?

Tip: For any point reflection, the distance between the original and the reflected point is always twice the distance from the original point to the line of reflection.