A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into a square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is Options.

We need to find the ratio of resistance between two endpoints of an edge in a triangle vs. a square, both made from identical wires of resistance R.

The wire of resistance R is bent into an equilateral triangle. Each side has resistance $$R/3$$.

Between two endpoints of one edge: one path has resistance $$R/3$$ (direct), and the other path has $$2R/3$$ (around the other two sides). These are in parallel:

$$R_{\triangle} = \frac{(R/3)(2R/3)}{R/3 + 2R/3} = \frac{2R^2/9}{R} = \frac{2R}{9}$$

The identical wire of resistance R is bent into a square. Each side has resistance $$R/4$$.

Between two endpoints of one edge: one path has $$R/4$$ (direct), and the other has $$3R/4$$ (around three sides). In parallel:

$$R_{\square} = \frac{(R/4)(3R/4)}{R/4 + 3R/4} = \frac{3R^2/16}{R} = \frac{3R}{16}$$

$$\frac{R_{\triangle}}{R_{\square}} = \frac{2R/9}{3R/16} = \frac{2}{9} \times \frac{16}{3} = \frac{32}{27}$$

The correct answer is Option (3): 32/27.