Why does the Pythagorean Theorem not work on this problem in the way that I used it? To begin with, I apologize for the vagueness of my question. It's hard to explain what exactly my question entails without seeing what process I went through to try to solve the problem. My question is just that I don't understand why my method did not work.
The problem
In Figure, $\mathrm{P}$ is a point in the square of side-length $10$ such that it is equally distant from two consecutive vertices and from the opposite side $\mathrm{AD}$. What is the length of $\mathrm{BP}$?
(A) 5
(B) 5.25
(C) 5.78
(D) 6.25
(E) 7.07
(I apologize for the crude drawing, the problem was in my book so I had to improvise using Paint.)
Figure
What I did: Since $\mathrm{BC}$ and $\mathrm{CD}$ are both $10$, I used the Pythagorean Theorem to get the length of diagonal $\mathrm{BD}$ as $\sqrt{200}$ and divide it by $2$. My answer was therefore (E) 7.07.
What my book did: Let $\mathrm{T}$ be the midpoint of $\mathrm{AB}$. Set $\mathrm{BP}$ to $x$, and the length of $\mathrm{BT}$ to $10-x$. To complete the triangle, they set the length of $\mathrm{PT}$ to $5$. Then they used the Pythagorean Theorem to do $x^2 = (10-x)^2 + 5^2$, yielding an answer of (D) 6.25.
While I understand how they did it, I simply cannot understand why my method didn't work. Is there some law that I'm not aware of pertaining to this problem? Since my incorrect answer was an answer choice, I assume there is a common error I'm making that was set as a trap.
Could someone explain this to me? Thank you very much.

The diagram in correct proportion. To get the square edge length $10,$ multiply all lengths by $$ \frac{10}{8} =  \frac{5}{4} = 1.25  $$

