If there are are 8 players, the circumference of the circle is divided into 8 equal arcs. Each one measures 360/8 = 45 degrees.
If there are are 9 players, the circumference of the circle is divided into 9 equal arcs. Each one is 360/9 = 40 degrees.
If there are are n players, the circumference of the circle is divided into n equal arcs. Each arc is 360/n degrees.
An angle is inscribed in a circle if the vertex lies on the circumference of the circle.
The measure of this inscribed angle is half the measure of the arc subtended by the angle.
Q: Find, with proof, the angle of the fourth exercise (throwing the ball to the fourth person on the left) with 9 players.)
9 players. 9 equal arcs of 40 degrees each.
The ball is thrown to the 4th player, then to the next 4th. That means the last player to catch the ball is the 8th player from Alice. That means there is still one more 40-degree arc before Alice, or, between the 8th player and Alice, there is one 40-degree arc.
This last 40-degree arc is the arc subtended by the angle in question.
Therefore, the angle of the fourth exercise is 40/2 = 20 degrees. ----answer.
Q: At one training session the coach suggests an exercise with an angle of 17.5 degrees. Alice protests that too many players would be needed. What is the smallest number of players required, and which exercise uses an angle of 17.5 degrees?
So, 17.5 *2 = 35 degrees ---the measure of the arc subtended by the angle of the exercise.
360/n = 35
360 = 35*n
n = 360/35 = 10.286 ---not an integer.
That means, the arc subtended by the angle in question is not one equal arc.
The arc subtended by the 17.5 degrees, which is 35 degrees, is a combination of more than one equal spaces between players.
How many equal spaces can there be in a 35-degree arc?
>>>Seven of 5-degree spaces.
>>>Five of 7-degree spaces.
360/7 = 51.428 players ---cannot be.
360/5 = 72 players ---could be.
Hence, there are 72 players,and each is separated by a 5-degree arc.
The desired angle, 17.5 degrees, needs 35 degrees, and that is the total arc for 7 of 5-degree spaces.
Umm, there is no way to throw the ball to two players and end up with 7 equal spaces left.
72 - 7 = 65
65/2 = 32.5 ---there is no player between the 32nd and 33rd players.
If we multiply that 32.5 by 2, we get 65.
....that is it!
Then we divide the 5-degree space by 2. We get 2.5 degrees per space.
Then, 360/2.5 = 144 equal spaces = 144 players.
Alice throws the ball to the 65th, then the 65th throws the ball to the, (65*2), 130th player.
144 -130 = 14 spaces left.
14*(2.5 degrees) = 35 degrees
The inscribed angle subtended by this 35-degree arc is
35/2 = 17.5 degrees ---the desired angle for the exercise.
Therefore, for this exercise,
>>>144 is the smallest number of players.
>>>throwing to the 65th player to the left is the pattern.