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Tick Tock Triangle

Problem

The hour hand on a wall clock is 3 inches long and the minute hand is 4 inches long. At three o"clock, the hands form a 3-4-5 right angle triangle.


At a random time during the day it is noted that the triangle formed by the hands has integer length sides. What is the probability that the triangle is isosceles?


Solution

Due to symmetry we can consider the hour hand to be on the right side of the clock. There is additional symmetry with the minute hand being the same distance before and after the hour hand, so we shall consider the cases where the minute hand is after the hour hand.

At 12 o"clock the minute and hour hand coincide and the distance between them is 1 inch. Similarly, at 6 o"clock, the distance is 7 inches. In both these cases the points are collinear, so a triangle is not formed. So there are precisely five possible integer distances between the ends of the minute and hour hand: 2, 3, 4, 5, and 6; the triangle will be isoscles when the distance is 3 or 4.

Therefore, P(isosceles|integer distance)=2/5.

At precisely what times does this happen?
It is interesting that P(integer distance)=0, P(distance 3 or 4)=0, but
P(isosceles|integer distance)=P(3 or 4)/P(integer distance)=2/5. How do you account for this?
Given that the length of the minute hand is m and the length of the hour hand is h, find P(isosceles|integer distance).
What about P(right-angle|integer distance)?

Problem ID: 167 (Apr 2004)     Difficulty: 3 Star

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