Circles, Permutations and Combinations

Karl plans to construct a circular swimming pool with a radius of 4 meters (see the green circle below). Two sprinklers A and B are installed on the edge of the pool. A water source (Point D) is located 14 meters from the center of the pool (point C). Two lines of pipes, which is tangent to the pool, will be connected to the two sprinklers.

Question 1 a1. What is the measure of ∠𝐷𝐴𝐶?

Answer explanation

Since the tangent of the circle is always perpendicular to the radius of the circle at the point of tangency, and perpendicular segments intersect at a right angle, then ∠𝐷𝐴𝐶=90 degree.

Question 1 a2. How long is the pipe which connects the water source and sprinkler A, i.e., AD?

• 14.426 𝑚

• 15.436 𝑚

• 12.436 𝑚

• 13.416 𝑚

Question 1 a3. How long is the pipe which connects the water source and sprinkler B, i.e., BD?

• 12.316 𝑚.

• 14.415 𝑚.

• 13.416 𝑚.

• 15.421 𝑚.

Answer explanation

Triangles ADC and BDC are congruent as both have a right angle, equal hypotenuse length and one other side of both triangles is also equal. CD (hypotenuse) is common to both triangles, and AC = BC has both are radii of the same circle.

Therefore A𝐷=𝐵𝐷,

hence 𝐵𝐷=𝐴𝐷=√180=13.416𝑚.

Question 1 a4. Suppose the distance of two sprinklers is 7.65 m. Karl decided to construct a fountain at point F (midway between two sprinklers) and another pipe must be installed to connect it from the water source. How far is fountain from the water source?

• 18.819 meters

• 12.859 meters

• 15.819 meters

• 19.829 meters

Question 1 a5. Find another (alternative) approach to solve this question of finding the length of DF?

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Question 2 a1. A basketball team consist of 5 players, although some additional players, called reserves, can be on the bench and used as replacements. By allocating alphabetical seven names (A, B, C, D, E, F, G) to the players, list all possible five-person team compositions that would be available to take the basketball court.

Answer explanation

(ABCDE) (ABCDF) (ABCDG) etc should be looking to see some systematic way that the list is being developed to ensure all combinations (n=21) have been found.

Question 2 b1. Coach John has only 8 players for his basketball team because some members of the team are not available. To start a game, he needs to submit the names of his 5 players to the organizing officials. Determine how many different teams of five players can be selected from the available players.

• There are 65 ways of selecting 5 players from 8 available players.

• There are 56 ways of selecting 5 players from 8 available players.

• There are 45 ways of selecting 5 players from 8 available players.

• There are 54 ways of selecting 5 players from 8 available players.

Answer explanation

Question 2 b2. When the coach considered the eight players available, he noted that three players performed best when they took up a ‘guard’ position and another three performed best in the role of ‘power forwards’. The remaining two players had the necessary skills to take on the role of ‘center’. The coach decided that the team should comprise two ‘guards’ and two ‘power forwards’ and one ‘center’. To select the strongest team, the players would be selected based on their strengths. On these criteria: how many possible teams of five players can be selected?

• 18 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑛𝑔 5 𝑝𝑙𝑎𝑦𝑒𝑟𝑠.

• 19 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑛𝑔 5 𝑝𝑙𝑎𝑦𝑒𝑟𝑠.

• 20 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑛𝑔 5 𝑝𝑙𝑎𝑦𝑒𝑟𝑠.

• 21 𝑤𝑎𝑦𝑠 𝑜𝑓 𝑠𝑒𝑙𝑒𝑐𝑡𝑖𝑛𝑔 5 𝑝𝑙𝑎𝑦𝑒𝑟𝑠.

Answer explanation