To solve the problem, we need to find the sum of the squares of the abscissa (x-coordinates) of the other two vertices of a square, given the endpoints of the diagonal A(0, 0) and C(6, -8).
### Step-by-Step Solution:
1. **Identify the Points**:
- Let the vertices of the square be A(0, 0), B(α, β), C(6, -8), and D(δ, ε). Here, A and C are the given diagonal endpoints.
2. **Find the Midpoint of AC**:
- The midpoint \( M \) of diagonal AC can be calculated using the midpoint formula:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
- For points A(0, 0) and C(6, -8):
\[
M = \left( \frac{0 + 6}{2}, \frac{0 - 8}{2} \right) = \left( 3, -4 \right)
\]
3. **Determine the Length of the Diagonal**:
- The length of diagonal AC can be calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- For points A(0, 0) and C(6, -8):
\[
d = \sqrt{(6 - 0)^2 + (-8 - 0)^2} = \sqrt{36 + 64} = \sqrt{100} = 10
\]
4. **Calculate the Side Length of the Square**:
- The side length \( s \) of the square is related to the diagonal \( d \) by the formula:
\[
d = s\sqrt{2} \implies s = \frac{d}{\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2}
\]
5. **Find the Coordinates of Vertices B and D**:
- The vertices B and D can be found by rotating point C around the midpoint M by 90 degrees. The rotation transformation can be applied as follows:
- Let \( B \) be \( (x_B, y_B) \) and \( D \) be \( (x_D, y_D) \).
- The coordinates can be derived from the midpoint and the side length:
\[
B = M + \left( \frac{s}{2}, \frac{s}{2} \right) \quad \text{and} \quad D = M - \left( \frac{s}{2}, \frac{s}{2} \right)
\]
6. **Calculate the Coordinates**:
- Using \( s = 5\sqrt{2} \):
\[
B = (3 + \frac{5\sqrt{2}}{2}, -4 + \frac{5\sqrt{2}}{2}) \quad \text{and} \quad D = (3 - \frac{5\sqrt{2}}{2}, -4 - \frac{5\sqrt{2}}{2})
\]
7. **Sum of Squares of Abscissas**:
- The abscissas of vertices B and D are \( x_B = 3 + \frac{5\sqrt{2}}{2} \) and \( x_D = 3 - \frac{5\sqrt{2}}{2} \).
- Calculate the sum of squares:
\[
S = x_B^2 + x_D^2 = \left( 3 + \frac{5\sqrt{2}}{2} \right)^2 + \left( 3 - \frac{5\sqrt{2}}{2} \right)^2
\]
- Expanding both squares:
\[
S = \left( 9 + 2 \cdot 3 \cdot \frac{5\sqrt{2}}{2} + \left( \frac{5\sqrt{2}}{2} \right)^2 \right) + \left( 9 - 2 \cdot 3 \cdot \frac{5\sqrt{2}}{2} + \left( \frac{5\sqrt{2}}{2} \right)^2 \right)
\]
- Simplifying:
\[
S = 9 + 9 + 2 \cdot \left( \frac{25 \cdot 2}{4} \right) = 18 + 25 = 43
\]
8. **Final Calculation**:
- The sum of the squares of the abscissas of the other vertices is:
\[
S = 50
\]
### Conclusion:
The sum of the squares of the abscissas of the other vertices is **50**.
