The graph of $ x^2 \plus{} y \equal{} 10$ and the graph of $ x \plus{} y \equal{} 10$ meet in two points. The distance between these two points is:

$ \textbf{(A)}\ \text{less than 1} \qquad 
\textbf{(B)}\ 1\qquad 
\textbf{(C)}\ \sqrt{2}\qquad 
\textbf{(D)}\ 2\qquad 
\textbf{(E)}\ \text{more than 2}$
1. We start with the given equations:
   \[
   x^2 + y = 10
   \]
   \[
   x + y = 10
   \]

2. To find the points of intersection, we solve these equations simultaneously. First, solve the linear equation for \( y \):
   \[
   y = 10 - x
   \]

3. Substitute \( y = 10 - x \) into the quadratic equation:
   \[
   x^2 + (10 - x) = 10
   \]

4. Simplify the equation:
   \[
   x^2 + 10 - x = 10
   \]
   \[
   x^2 - x = 0
   \]

5. Factor the quadratic equation:
   \[
   x(x - 1) = 0
   \]

6. Solve for \( x \):
   \[
   x = 0 \quad \text{or} \quad x = 1
   \]

7. Substitute \( x = 0 \) and \( x = 1 \) back into the linear equation \( y = 10 - x \) to find the corresponding \( y \)-values:
   - For \( x = 0 \):
     \[
     y = 10 - 0 = 10
     \]
     So, one point of intersection is \( (0, 10) \).

   - For \( x = 1 \):
     \[
     y = 10 - 1 = 9
     \]
     So, the other point of intersection is \( (1, 9) \).

8. Now, we calculate the distance between the points \( (0, 10) \) and \( (1, 9) \) using the distance formula:
   \[
   \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]
   Substituting the coordinates:
   \[
   \text{Distance} = \sqrt{(1 - 0)^2 + (9 - 10)^2}
   \]
   \[
   \text{Distance} = \sqrt{1^2 + (-1)^2}
   \]
   \[
   \text{Distance} = \sqrt{1 + 1}
   \]
   \[
   \text{Distance} = \sqrt{2}
   \]

The final answer is \(\boxed{\sqrt{2}}\).