The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of me

Let the place that they touch be x meters from the 300 m wall

Then the distance from the 500 m wall will be 1200 - x

Then....the total distance, D, can be expressed as

D = sqrt [ x^2 + 300^2 ] + sqrt [ (1200 - x)^2 + 500^2 ]

This can be solved with Calculus, but it's a little messy

A graph seems better : 

https://www.desmos.com/calculator/jwjzdpvz1b

The graph shows that the minimum distance is achieved when x = 450m

And the minimum distance  ≈  1442 m

By the way......we can show that this is true if

arctan [ 300 / 450]   =  arctan [ 500 / 750 ]  ......and this is true  !!!!