To find the [tex]\( y \)[/tex]-coordinate of the other endpoint of the diameter, let's carefully analyze the given information about the circle.
Given:
- The equation of the circle is [tex]\((x-7)^2 + (y-8)^2 = 10\)[/tex].
- The center of the circle is [tex]\((7, 8)\)[/tex].
- The radius of the circle is [tex]\(\sqrt{10}\)[/tex].
- One endpoint of the diameter has [tex]\( y \)[/tex]-coordinate [tex]\(11\)[/tex].
### Step-by-Step Solution
1. Identify the center's [tex]\( y \)[/tex]-coordinate:
The center of the circle is at [tex]\((7, 8)\)[/tex]. This means the [tex]\( y \)[/tex]-coordinate of the center is [tex]\( 8 \)[/tex].
2. Determine the given [tex]\( y \)[/tex]-coordinate of one endpoint of the diameter:
One endpoint of the diameter has a [tex]\( y \)[/tex]-coordinate of [tex]\( 11 \)[/tex].
3. Calculate the vertical distance between the center and the given endpoint:
The distance (or difference) between the [tex]\( y \)[/tex]-coordinate of the center (which is [tex]\( 8 \)[/tex]) and the given endpoint (which is [tex]\( 11 \)[/tex]) can be found by:
[tex]\[
\text{Difference} = 11 - 8 = 3
\][/tex]
4. Use symmetry to find the [tex]\( y \)[/tex]-coordinate of the other endpoint:
Since the given endpoint is above the center by [tex]\( 3 \)[/tex] units, the other endpoint will be symmetrically placed below the center by the same distance. Therefore:
[tex]\[
\text{Other } y \text{-coordinate} = 8 - 3 = 5
\][/tex]
Hence, the [tex]\( y \)[/tex]-coordinate of the other endpoint of the diameter is [tex]\( \boxed{5} \)[/tex].
