An electricity and magnetism problem by S M

Let ABCDA be the circle, with A & C diametrically opposite, and the same case with B & D. Draw the circle with this description, for convenience.

We need to calculate the resistance of this loop between A and B. The wires ADC and ABC will have resistances 5ohm each. Hence half of these wires will have 2.5ohm each. Hence, the equivalent resistance will be equal to

${ R }_{ equivalent\quad }=\frac { 2.5\Omega \quad X\quad 7.5\Omega }{ 2.5\Omega \quad +\quad 7.5\Omega } =\quad 1.875\Omega$

Since the two points form $90^\circ$ at the center, we see that the lengths of the wires are in the ratio $3:1$ . These wires are in parallel since they are between the same Potentials .

The total resistance is given to be $10\Omega$ . So , the resistances of the two wires are $\frac{10}{4}\Omega$ and $\frac{30}{4}\Omega$ respectively.

Since they are in parallel , $Equivalent Resistance$ = $\frac{0.75 * 2.5 }{2.5 + 7.5}$ = $\boxed{1.875\Omega}$