Tape winded around rectangular board

As shown in a figure, there is a translucent red tape in which the end is cut at 45 degrees and there is a line drawn in the center of the tape. Moreover, as shown in a figure, there is a rectangular transparent board and the point P is set on the side BC so as to BP = 3cm.

As shown in Fig. 1, the end of the central line of the tape is put on the point P and the tape is winded around a board on the surface.

The central line of the tape crosses each side of the rectangle at 45 degrees.

As for the sample in Fig.1, the tape is winded to the point S and cut with scissors along the side.

In this case, the length of the central line of the winded tape becomes PQ + QR + RS.

In addition, the shadow area of Fig. 1 is a portion where the surface and the back of the board are covered by the tape and look deep red.

Fig.2 shows that the tape is winded partially to the middle in case of the rectangle with AB=4cm and AD=12cm.

Then, the tape continues to be winded further and when the central line of the tape overlaps with the point P again, it is stopped winding and the tape is cut with scissors along the side.

Answer the following questions about the case of Fig. 2.

Consider if necessary that in case the length of two sides forming a right angle of a right-angled isosceles triangle is 1 cm, the length of the 3rd side of the triangle is assumed to be 1.414 cm.

Moreover, the thickness of the tape and the board is assumed to be 0 cm.

Answer

(1) Write the central line of the tape winded around in the figure. Noted that write the central line of the tape on the back with a dotted line.

(2) Find the length of the central line of the winded tape.

(3) Find the area of the portion covered on the tape among the surfaces of a board.

(4) Find the area of the portion which is covered by the tape both of the surface and the back of the board and looks deep red.