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 $b$?

The sum of a and b is 12.

Step-by-step explanation:

Try drawing this figure. ABCD would actually be a triangle, rather than the common quadrilateral you would expect -  considering there are 4 points plotted. Therefore, we would have to find the distance between respective points ( 0, 1 ) and ( 2, 5 ) / ( 2, 5 ) and ( 7, 0 ) / ( 0, 1 ) and ( 7, 0 ). Let's apply the distance formula and calculate the distance between each point, therefore determining the perimeter.

Distance between points ( 0, 1 ) and ( 2, 5 )

=

=

=

= =

Distance between points ( 2, 5 ) and ( 7, 0 )

=

=

=

= =

And you can calculate the distance between points ( 0, 1 ) and ( 7, 0 ) to be exactly as well. Therefore, you can say that the perimeter of this triangle is = . Note that this is similar to the form we are given. Thus, a = 10, and b = 2 - making it's sum 10 + 2 = 12.