# 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 8, 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 AD. What is the length of 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 8

What I did: Since BC and CD are both 10, I used the pythagorean theorem to get the length of diagonal BD (sqrt 200) and divided by 2. My answer was therefore E) 7.07.

What my book did: Set BP to x, and the length of (B and midpoint of AB) to 10-x. To complete the triangle, they set the length of (P and midpoint of AB) 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.

• It is because $BP$ does not lie on the diagonal of the square. In fact, if that was the case, that would be $10\sqrt{2} /2$ long, but $P$ would be the center of the square, making the third segment long $5$ and thus different from the other two. – Harnak Dec 27 '16 at 22:11
• @Harnak, was just going to say this. Nice. – The Count Dec 27 '16 at 22:12
• @Harnak OH that was very careless of me to assume that BP is the same length as PD. It makes sense now, thank you! – ak_27 Dec 27 '16 at 22:13
• @ak_27, it is awesome that you came to this website for help with this. most people would just get frustrated and give up. keep at it! i loved this question. very subtle. – The Count Dec 27 '16 at 22:14
• You're welcome ^^ – Harnak Dec 27 '16 at 22:15

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