Final answer:
The equation of the posts traced by the man on the racecourse can be found using the Pythagorean theorem. The possible positions of the flag posts are at a distance of 2 m and 9 m from the man.
Step-by-step explanation:
The equation of the posts traced by the man can be found using the concept of Pythagorean theorem. Let's assume that the man is standing at point A and the two flag posts are at points B and C. We know that the sum of the distances from the man to the two flag posts is always 10 m, and the distance between the flag posts is 8 m.
Using the Pythagorean theorem, we can write the following equation:
AB^2 + AC^2 = BC^2
Substituting the known values, we get:
AB^2 + AC^2 = (8)^2
Using the fact that AB + AC = 10, we can substitute AB = 10 - AC.:
(10 - AC)^2 + AC^2 = 64
Simplifying the equation:
100 - 20AC + AC^2 + AC^2 = 64
2AC^2 - 20AC + 36 = 0
Solving this quadratic equation, we find that AC = 2 and AC = 9. Hence, the possible positions of the flag posts are at a distance of 2 m and 9 m from the man.