Take an equilateral triangle and circumscribe a circle around it (like in the leftmost image). Then draw a random chord in the circle. What is the probability that the the chord is longer than a side of the triangle?
1. If you take the chord and rotate the triangle so that one of the corners is on one of the endpoints of the chord then if the other endpoint intersects the opposite side of the triangle (from the corner). Clearly by arclengths there is a 1/3 chance that the chord is longer than a side.
2. If instead you rotate the triangle so that it is parallel to the chord. And you add in a 2nd triangle and rotate it so that one of its sides is also parallel but (on the other side... harder to describe than to look at the 2nd image). If the chord then lies between the 2 it is longer, and if you take the diameter perpendicular to these lines then 1/2 of it lies between the lines and 1/2 outside of them. So there is a 1/2 chance that the chord is longer than a side.
3. However if you were to inscribe a circle inside the equilateral triangle (this will have radius 1/2 that of the larger triangle). Take your random chord and look at it's midpoint, if that midpoint falls inside the smaller circle than the chord is longer than a side of the triangle. And by areas there is 1/4 chance of this happening.
Therefore a random chord can have 1/3, 1/2, or 1/4 chance of being longer. I'm not sure how well I described this, and I would like to write a lesson based on it so if you can give me pointers that would be helpful.
1. If you take the chord and rotate the triangle so that one of the corners is on one of the endpoints of the chord then if the other endpoint intersects the opposite side of the triangle (from the corner). Clearly by arclengths there is a 1/3 chance that the chord is longer than a side.
2. If instead you rotate the triangle so that it is parallel to the chord. And you add in a 2nd triangle and rotate it so that one of its sides is also parallel but (on the other side... harder to describe than to look at the 2nd image). If the chord then lies between the 2 it is longer, and if you take the diameter perpendicular to these lines then 1/2 of it lies between the lines and 1/2 outside of them. So there is a 1/2 chance that the chord is longer than a side.
3. However if you were to inscribe a circle inside the equilateral triangle (this will have radius 1/2 that of the larger triangle). Take your random chord and look at it's midpoint, if that midpoint falls inside the smaller circle than the chord is longer than a side of the triangle. And by areas there is 1/4 chance of this happening.
Therefore a random chord can have 1/3, 1/2, or 1/4 chance of being longer. I'm not sure how well I described this, and I would like to write a lesson based on it so if you can give me pointers that would be helpful.
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