I am reading a book for beginner about building the chords, basically on triads so far. It is said that the major triads is formed by a root, third and a fifth. From that instruction, it is easily to setup a C major triad by making the root as C; to get the third, I add 4 half steps to C so to get E; to get the fifth, I add 7 half steps to C so to get G. The C major triad is shown on the left of the following stave
Similarly, I am getting the F major triad by using a F note as the root, A as the third and C as the fifth, so F-A-C be the F major triad, correct?
In the book, the author introduces anohter way to construct major triads by taking the first, major third and fifth notes from a major scale. It then gives me F major triad as shown in the right side of above stave. The book does not tell why there is a single-flat key signature added. I am trying to reason it by listing the C major scale, C-D-E-F-G-A-B, from that, the first, major third and fifth notes is just C-E-G but this does not give me F-A-C. However, I find that if I lower the F by half step, I get the E, which may get E-A-C by adding one-flat signature on F. This still does not show me why the RHS staff is connected to the F major triad.
Also, is it any noticeable difference between the F major triads with and without the flat key signature? I tried to repeat the following pattern in a virtual keyboard and listen very carefully, I can't tell the difference honestly
(I am trying to search the related questions on generating F mjaor triad but it comes up with few threads with some terms I don't understand yet).