Point Q lies inside rectangle ABCD so that QA = 2, QB = 3, and QC = 4.

Call the length and width of the rectangle L and W. Then when we draw the point Q inside the rectangle, we can connect Q to each vertex (to get the lines of length 2, 3 and 4, along with 'x', the length we want to find). If we then connect point Q to each side using perpendicular lines, we'll subdivide the length and width into two parts, and if we invent a letter, say "d", for one part of the length, the remaining part will be L-d. Similarly we can call the two resulting parts of the width "c" and "W - c". Then we can use Pythagoras several times to get to an answer. I illustrated how I divided up the diagram below.

So looking say at the top left corner of the diagram below, we find:

c^2 + d^2 = 3^2 = 9

and respectively, using the bottom right corner and the top left corner, we learn

d^2 + (W - c)^2 = 2^2 --> (W - c)^2 = 4 - d^2
c^2 + (L - d)^2 = 4^2 --> (L - d)^2 = 16 - c^2

Now we want to find x, and looking at the bottom right, we can see

x^2 = (L - d)^2 + (W - c)^2

and using the equations above, we can substitute for (L-d)^2 and for (W - c)^2 to get

x^2 = 16 - c^2 + 4 - d^2 = 20 - (c^2 + d^2)

and finally using the first Pythagorean equation above, we can substitute for c^2 + d^2 to get

x^2 = 20 - 9 = 11
x = √11