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Answers to the problem of playing billiards

The problem of finding the length of the billiards path that requires solving with 6th grade knowledge is not too difficult, but not many people can solve it.

Threads:

The figure on the left shows a player hitting a billiard ball against the side of the table with an angle of reflection equal to the angle of incidence and after three reflexes it hits another billiard ball.

The figure on the right shows a ball hit from corner A of a billiard table ABCD which is a rectangle with AD = 3, AB = 10 such that the ball hits and reflects at the first side CD. The ball bounces then continues to touch – reflect from the edges of the table with the angle of reflection equal to the angle of incidence and after five bounces the ball stops at angle D.

What is the total length that the ball has traveled from A to D?





Answers to the problem of playing billiards

Addendum: The Pythagorean Theorem and Leonardo da Vinci’s proof.

Pythagorean theorem: If triangle ABC is right-angled at A, then BC2 = AB2 + AC2

Leonardo da Vinci (1452-1519) was the almighty genius of mankind when he was a painter, architect, musician, doctor, engineer, sculptor, anatomist, inventor and philosopher.

Leonardo da Vinci’s proof of the Pythagorean theorem:

Constructing and dividing the figure, we see that the area of ​​the square built on the hypotenuse is equal to the sum of the areas of the two squares built on the two sides of the right angle or
BC2 = AB2 + AC2





The answer to the problem of playing billiards - 1

Apply:

The ball hits from A, hits the table at N then bounces and then continues to touch-reflex from the edges of the table with the angle of reflection equal to the angle of incidence and after 5 reflections the ball stops at angle D.





Answers to the problem of playing billiards - 2

Because the rectangle ABCD receives KE connecting the 2 midpoints of AD and BC as a symmetry axis, and at the same time, the starting point A and the ending point D of the billiards are 2 symmetrical points through KE, so in 5 billiards points touching the table has 2 pairs of points (M, N) and (P, Q) symmetrical through KE and the other touching point is E midpoint BC.

From this it follows that the shapes AMND, MNQP and PQCB are rectangles with AM = DN = NQ = MP = 4, PB = QC = 2 and KA = KD = EB = EC = 3/2 (see figure).

Using the Pythagorean theorem we have: AN = DM = NP = MQ = QE + PE = 5.

So the total length that the ball has traveled from A to D is
AN + NP + PE + EQ + QM + MD = 25.

Tran Phuong

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