Answer
The resistance of the two connected wires is $~~\frac{R}{4}$
Work Step by Step
We can write an expression for the resistance of the original wire:
$R = \frac{\rho~L}{A}$
We can write an expression for the resistance of each half of the wire after it has been cut:
$R' = \frac{\rho~(L/2)}{A}$
$R' = \frac{\rho~L}{2A}$
$R' = \frac{R}{2}$
We can find the resistance $R_{eq}$ of the connected wires:
$\frac{1}{R_{eq}} = \frac{1}{R/2} + \frac{1}{R/2}$
$\frac{1}{R_{eq}} = \frac{2}{R} + \frac{2}{R}$
$\frac{1}{R_{eq}} = \frac{4}{R}$
$R_{eq} = \frac{R}{4}$
The resistance of the two connected wires is $~~\frac{R}{4}$