Paul wants to install two zip lines between two pylons perpendicular to the ground, one 40 meters high and the other 30 meters high. the pylons are spaced 50 meters apart. Paul wants the zip lines to be the same length. how long are the zip lines

To find the length of the zip lines, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Let x be the length of the zip lines.

For the first zip line (40 meters high), the length of the zip line can be found by
\sqrt{40^2 + 50^2}

For the second zip line (30 meters high), the length of the zip line can be found by
\sqrt{30^2 + 50^2}

Since Paul wants the zip lines to be the same length, we set these two equations equal to each other:
\sqrt{40^2 + 50^2} = \sqrt{30^2 + 50^2}

Solving for x:
40^2 + 50^2 = 30^2 + 50^2
1600 = 900
x = \sqrt{900}
x = 30 \text{ meters}

\boxed{30 \text{ meters}} is the length of the zip lines.