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(D) To find the equivalent resistance between $A$ and $B$,we simplify the circuit step-by-step.$1$. The two resistors of resistance $r$ in the middle

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$\frac{8}{7}r$

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(D) To find the equivalent resistance between $A$ and $B$,we simplify the circuit step-by-step.$1$. The two resistors of resistance $r$ in the middle branch are in series,giving $r + r = 2r$.$2$. This $2r$ is in parallel with the top resistor $r$,giving equivalent resistance $R_p = \frac{r \cdot 2r}{r + 2r} = \frac{2}{3}r$.$3$. Now,the circuit consists of the left branch (resistance $r$),the right branch (resistance $r$),the top branch (resistance $\frac{2}{3}r$),and the bottom branch (resistance $r + r = 2r$ between $A$ and $B$).$4$. Simplifying further as shown in the diagram,the total resistance between $A$ and $B$ is the parallel combination of the upper path (total resistance $r + \frac{2}{3}r + r = \frac{8}{3}r$) and the lower path (total resistance $2r$).$5$. $R_{eq} = \frac{(\frac{8}{3}r) \cdot (2r)}{\frac{8}{3}r + 2r} = \frac{\frac{16}{3}r^2}{\frac{14}{3}r} = \frac{16}{14}r = \frac{8}{7}r$.

What is the equivalent resistance between points $A$ and $B$ in the given circuit?

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