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

$\sqrt{83}$

Explanation for the correct option:

Option (C): $\sqrt{\mathbf{83}}$

Applying the distance formula we get,

$\mathit{A}\mathit{B}\mathbf{=}\sqrt{{\left(a_1-b_1\right)}^{\mathbf{2}}\mathbf{+}{\left(a_2-b_2\right)}^{\mathbf{2}}}$

As the given co-ordinates must be integers,

$A{B}^2$must be the sum of two perfect squares.

$83$ cannot be expressed as the sum of two perfect squares.

Hence for integer co-ordinates $AB$ cannot be $\sqrt{83}$

Explanation for the incorrect option:

Option (A): $\sqrt{\mathbf{65}}$

$65=8^2+1^2$ ,$65$ can be expressed as sum of two perfect squares.

Option (B): $\sqrt{\mathbf{74}}$

$74=7^2+5^2$,$74$ can be expressed as sum of two perfect squares

Option (D): $\sqrt{\mathbf{97}}$

$97=9^2+4^2$,$97$ can be expressed as sum of two perfect squares

Hence for integer co-ordinates $AB$ cannot be $\sqrt{83}$

Hence option $\left(C\right)$ is the correct answer.