### Instruction: Segment $s_1$ has endpoints at $(3+\sqrt{2},5)$ and $(4,7)$.  Segment $s_2$ has endpoints at $(6-\sqrt{2},3)$ and $(3,5)$.  Find the midpoint of the segment with endpoints at the midpoints of $s_1$ and $s_2$.  Express your answer as $(a,b)$.### Response: To find the midpoint of a segment, I can use the formula that averages the x-coordinates and the y-coordinates of the endpoints.
So, to find the midpoint of $s_1$, I need to average $3+\sqrt{2}$ and $4$, and also average $5$ and $7$.
That gives me $\left(\frac{7+\sqrt{2}}{2},6\right)$ as the midpoint of $s_1$.
Similarly, to find the midpoint of $s_2$, I need to average $6-\sqrt{2}$ and $3$, and also average $3$ and $5$.
That gives me $\left(\frac{9-\sqrt{2}}{2},4\right)$ as the midpoint of $s_2$.
Now, to find the midpoint of the segment with endpoints at the midpoints of $s_1$ and $s_2$, I need to apply the same formula again.
This time, I need to average $\frac{7+\sqrt{2}}{2}$ and $\frac{9-\sqrt{2}}{2}$, and also average $6$ and $4$.
To simplify the fractions, I can combine the numerators using the distributive property of multiplication over addition.
That gives me $\frac{16}{2}$ as the average of the x-coordinates, and $5$ as the average of the y-coordinates.
So, the final answer is $\left(8,5\right)$.