Treat the rise and run as proportional to $5$ and $12$ to form a smaller similar right triangle.

Use the Pythagorean theorem with $5$ and $12$ to determine the hypotenuse of the similar triangle.

Scale the similar triangle to match hypotenuse $26$, then use $\sin A=\frac{\text{opposite}}{\text{hypotenuse}}$.

Enter 26/13 to get the scale factor, then enter 5*(26/13) to get the rise.

Enter (5*(26/13))/26 to compute $\sin A$ as a decimal.

Convert the decimal to a fraction (or simplify the ratio by hand) and select the matching option.

A rise of 5 feet for every 12 feet of run means the vertical leg and horizontal leg are in the ratio $5:12$.

A right triangle with legs $5$ and $12$ has hypotenuse $13$ because $5^2+12^2=13^2$.

Since the actual hypotenuse is $26$, the scale factor is $\frac{26}{13}=2$, so the actual rise is $5\cdot 2=10$.

Angle $A$ is at the bottom, so the side opposite $A$ is the rise, and the hypotenuse is the ramp length. Thus,

So the correct choice is $\dfrac{5}{13}$.