Answer: 35

 

Solution:

 

 

First, I drew a diagram illustrating the problem.

 

 

 

 

Note, this diagram isn't necessarily to scale, because I don't know
how big x is to begin with.

 

 

Let s be the length of each side of square S.

 

 

Let the square be ABCD, so AP=7, BP=35, CP=49, and DP=x.

 

 

Drop altitudes from P to each side of S, and label the points as follows:

 

 

 

 

E is on AB, F is on BC, G is on CD, and H is on DA.

 

 

Label the lengths of the segments as follows:

 

 

EP, AH, and BF have length a; GP, DH, and CF have length s-a

FP, BE, and CG have length b; HP, AE, and DG have length s-b

 

 

By the Pythagorean Theorem, we have

 

 

EPB: a²+b²=35²

FPC: (s-a)²+b²=49²

EPA: a²+(s-b)² =7²

DPH: (s-a)²+(s-b)²=x²

 

 

By subtracting the first and fourth of these equations from the sum of the second and third, we get

 

 

0 = 49² + 7² - 35² - x², or

x² = 49² + 7² - 35²

x² = 35²

x = 35

 

 

Since DP and BP are equal, it follows that P is on the diagonal AC, so
the diagram is just a little misleading (which is why I didn't include it
in the original statement of the problem).