Question

Let $A=(a_1,a_2)$ and $B=(b_1,b_2)$ be two points in the plane with integer coordinates. Which one of the following is not a possible value of the distance between A and B?

• A. $\sqrt{65}$
• B. $\sqrt{74}$
• C. $\sqrt{83}$
• D. $\sqrt{97}$

Answer 1

The correct option is C $\sqrt{83}$

Distance between $A$ and $B$ is =$\sqrt{(a_1-b_1{)}^2+(a_2-b_2{)}^2}$

It's given that the coordinates are integer, therefore $(a_1-b_1{)}^2$ and $(a_2-b_2{)}^2$ are the squares of integer numbers.

In options, we have

$\sqrt{64}=\sqrt{8^2+1^2}$

$\sqrt{74}=\sqrt{7^2+5^2}$

$\sqrt{97}=\sqrt{9^2+4^2}$

$\sqrt{83}$ cannot be written in this form.

Therefore, option C cannot the distance between A and B.