Consider the fourteen lines in the plane given by $y = x + r,\ y = - x + r,\$ where $r \in \{ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6\}$. The number of squares formed by these lines, whose sides are of length $\sqrt{2}$, is

Question

Dr. Abubacker · Accepted Answer

(c)
The given lines are perpendicular to each other.
$	herefore Perpendicular\ distance = \frac{\left| r_{1} - r_{2} ight|}{\sqrt{2}} = \sqrt{2}$
$\Rightarrow \ r_{1} - r_{2} = 2$
The difference between the $y$-intercepts $= 2$
This can happen for five combinations {(0, 2), (1, 3), (2, 4), (3, 5), (4, 6)}.
The difference between the $x$-intercepts $= 2$
This can happen for five combinations.
Hence, total number of squares $= 5 	imes 5 = 25$

Q. Consider the fourteen lines in the plane given by $y = x + r,\ y = - x + r,\$ where $r \in \{ 0,\ 1,\ 2,\ 3,\ 4,\ 5,\ 6\}$. The number of squares formed by these lines, whose sides are of length $\sqrt{2}$, is

Explanation

Reviewed by Dr. Abubacker

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