Let $A=(0,1),$ $B=(2,5),$ $C=(5,2),$ and $D=(7,0).$ A figure is created by connecting $A$ to $B,$ $B$ to $C,$ $C$ to $D,$ and $D$ to $A.$ The perimeter of $ABCD$ can be expressed in the form $a\sqrt2+b\sqrt{5}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$?
We use the distance formula to find the length of each side.

The distance from $(0, 1)$ to $(2, 5)$ is $\sqrt{(2 - 0)^2 + (5 - 1)^2} = 2\sqrt{5}$.

The distance from $(2, 5)$ to $(5, 2)$ is $\sqrt{(5 - 2)^2 + (2 - 5)^2} = 3\sqrt{2}$.

The distance from $(5, 2)$ to $(7, 0)$ is $\sqrt{(7 - 5)^2 + (0 - 2)^2} = 2\sqrt{2}$.

The distance from $(7, 0)$ to $(0, 1)$ is $\sqrt{(0 - 7)^2 + (1 - 0)^2} = 5\sqrt{2}$.

Adding all of these side lengths, we find that the perimeter is $10\sqrt{2} + 2\sqrt{5}$. Thus, our final answer is $10 + 2 = \boxed{12}$.
12
Level 4
Algebra
Problem: Let $A=(0,1),$ $B=(2,5),$ $C=(5,2),$ and $D=(7,0).$ A figure is created by connecting $A$ to $B,$ $B$ to $C,$ $C$ to $D,$ and $D$ to $A.$ The perimeter of $ABCD$ can be expressed in the form $a\sqrt2+b\sqrt{5}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the perimeter of the quadrilateral formed by points A, B, C, and D, we first need to calculate the lengths of the sides AB, BC, CD, and DA. 

1. Calculate the distance AB:
   A = (0, 1), B = (2, 5)
   Distance AB = sqrt((2 - 0)^2 + (5 - 1)^2) = sqrt(2^2 + 4^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5).

2. Calculate the distance BC:
   B = (2, 5), C = (5, 2)
   Distance BC = sqrt((5 - 2)^2 + (2 - 5)^2) = sqrt(3^2 + (-3)^2) = sqrt(9 + 9) = sqrt(18) = 3sqrt(2).

3. Calculate the distance CD:
   C = (5, 2), D = (7, 0)
   Distance CD = sqrt((7 - 5)^2 + (0 - 2)^2) = sqrt(2^2 + (-2)^2) = sqrt(4 + 4) = sqrt(8) = 2sqrt(2).

4. Calculate the distance DA:
   D = (7, 0), A = (0, 1)
   Distance DA = sqrt((0 - 7)^2 + (1 - 0)^2) = sqrt((-7)^2 + 1^2) = sqrt(49 + 1) = sqrt(50) = 5sqrt(2).

Now, sum the lengths of all sides to find the perimeter:
Perimeter = AB + BC + CD + DA = 2sqrt(5) + 3sqrt(2) + 2sqrt(2) + 5sqrt(2).

Combine like terms:
Perimeter = 2sqrt(5) + (3 + 2 + 5)sqrt(2) = 2sqrt(5) + 10sqrt(2).

The perimeter can be expressed in the form a√2 + b√5, where a = 10 and b = 2. 

Finally, calculate the sum of a and b:
Sum = a + b = 10 + 2 = 12. 

Thus, the final answer is 12.<|END_OF_LATENT|> The final answer is $12$.