This is a right triangle since

See diagram:

The side of 5 is the geometric mean between x and 13.

The side 12 is the geometric mean between y and 13.

The length of the perpendicular (h) to the hypotenuse is the geometric mean between x and y.

Can you finish up.

Sorry, I misread your instructions. I did not use Heron's formula to solve this.....I'll see if someone else does. Otherwise, I'll be back.

Okay, I'm back. I really don't know how Heron's formula fits into all this. Heron's formula is a formula for finding the area of any triangle knowing only the lengths of the sides.

, where a, b, and c are the sides and s is the semi-perimeter. That is,

The perimeter of your triangle = 5 + 12 + 13 = 30.

The semi-perimeter (s) = 15

Substituting into Heron's formula to find Area:

This could just as easily been found using using b = 12 and h = 5. But, I digress.

Now, since you just found what the area was, you can use this to solve for your missing perpendicular.

All done!!