How many times does the rotation r (45°, P) need to be applied to a figure to map the figure to itself?

8 times.

Step-by-step explanation:

The rotaton is defined as r (45°, P).

So, the rotation is 45°. Now, to map the figure onto itself it needs to rotate 360°. If the rotation is 45°, then we need 8 times that rotation to get 360°, because

45° x 8 = 360°

Therefore, there are need 8 rotation to map the figure onto itself.

9 times cause 9 around the clock times 5 is 45