Point P leaves vertex A and moves straight the inside of the square of one side of 80cm according to the next rule.
< Rule >
・ When P hits each side, it rebounds.
・ When P hits each vertex, it stops.
As shown in the Fig.1 below, the angle point P hits is set to X and the angle P rebounds is set to Y.
Answer to the following questions.
(1) Y is equal to X and point P rebounds at the position of 30 cm from vertex B.
Which is the vertex that is the nearest to the position where P rebounds at the fourth time?
In addition, find the distance from the vertex.
(2) In the case of (1), how many times does P rebound before it stops?
(3) Y is half of X. The locus before P rebounds at fourth time became as shown in the Fig.2 below.
Find the angle of Z.
Answer
(1) C, 10 cm
(2) 9 times
(3) 64 degrees
Solution
(1) P rebounds as shown in Fig.3.
The position rebounds at the fourth time is P4 and the near vertex is C.
Because P1P4 = AP2 = 60cm, CP4 = (30 + 60) - 80 = 10 cm.
(2) According to 60 × 4 = 80 × 3, the locus is as shown in Fig.4.
It is nine times in all.
(3) When determining angles from Q to W as shown in Fig.5 below, ∠ R= ∠ Z/2,
< Rule >
・ When P hits each side, it rebounds.
・ When P hits each vertex, it stops.
As shown in the Fig.1 below, the angle point P hits is set to X and the angle P rebounds is set to Y.
Answer to the following questions.
(1) Y is equal to X and point P rebounds at the position of 30 cm from vertex B.
Which is the vertex that is the nearest to the position where P rebounds at the fourth time?
In addition, find the distance from the vertex.
(2) In the case of (1), how many times does P rebound before it stops?
(3) Y is half of X. The locus before P rebounds at fourth time became as shown in the Fig.2 below.
Find the angle of Z.
Answer
(1) C, 10 cm
(2) 9 times
(3) 64 degrees
Solution
(1) P rebounds as shown in Fig.3.
The position rebounds at the fourth time is P4 and the near vertex is C.
Because P1P4 = AP2 = 60cm, CP4 = (30 + 60) - 80 = 10 cm.
(2) According to 60 × 4 = 80 × 3, the locus is as shown in Fig.4.
It is nine times in all.
(3) When determining angles from Q to W as shown in Fig.5 below, ∠ R= ∠ Z/2,
∠ S =90 ° - Z/2
∠ T = ∠ S/2 = 45 ° - Z/4
∠ U = 90 ° - ∠ T = 45 ° + Z/4
∠ V = ∠ U/2 = 22.5 ° + Z/8
∠ W = ∠ Z
According to ∠ W + ∠V + 85.5 ° = 180°,
Z + 22.5 + Z/8 + 85.5 = 180, then
Z + Z/8 = 72
Therefore Z = 64 °
∠ T = ∠ S/2 = 45 ° - Z/4
∠ U = 90 ° - ∠ T = 45 ° + Z/4
∠ V = ∠ U/2 = 22.5 ° + Z/8
∠ W = ∠ Z
According to ∠ W + ∠V + 85.5 ° = 180°,
Z + 22.5 + Z/8 + 85.5 = 180, then
Z + Z/8 = 72
Therefore Z = 64 °
