10. (15 points) A turtle and a rabbit are in a 1000-meter race. The rabbit's speed is 5 times that of the turtle. When they start from the starting line at the same time, the turtle runs continuously, while the rabbit runs to a certain point and then starts sleeping. When the rabbit wakes up, the turtle has already taken the lead. The rabbit then tries to catch up, but when the turtle reaches the finish line, the rabbit is still 10 meters behind. How many meters did the turtle run while the rabbit was sleeping?
【Analysis】First, consider the total distance the rabbit would run without sleeping as 990 meters. The distance the tortoise runs, and the remaining distance is the distance the tortoise runs while the rabbit is sleeping.

【Solution】First, according to the rabbit's speed being 5 times that of the tortoise, the distance the rabbit runs is 5 times that of the tortoise. When neither rests, the rabbit runs the full distance of $1000-10=990$ (meters); the tortoise runs a distance of $990 \div 5=198$ (meters);

The distance the tortoise runs while the rabbit is sleeping is: $1000-198=802$ (meters)
Answer: The tortoise runs 802 meters while the rabbit is sleeping.