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B is a point in the top circle of a right circular cylinder. C is a point in the bottom circle of the given cylinder. The angle between [BC] and the base's plan of the cylinder is 45 degrees. The radius of the cylinder is 25cm and [BC] is 14√2 cm.

Find the distance between the axis of this cylinder and the plan formed from the segment [BC] and which is parallel with the axis.

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  • $\begingroup$ Impossible to determine with the information at hand. These constraints don’t prevent $BC$ from intersecting the axis, for instance. $\endgroup$ – amd Jun 14 at 20:40
  • $\begingroup$ @amd I'm not expert in Geometry but, I think that when it says the plan which is defined by [BC] is parallel with the axis, this means they don't intersect with one another. $\endgroup$ – Dionis Beqiraj Jun 14 at 20:55
  • $\begingroup$ My point is that $B$ and $C$ are not constrained enough by the problem statement to allow a unique solution: the length of the projection of $BC$ onto the base is much less than the diameter of the base. From this intersecting position, shift both points a small distance in the same direction. The resulting plane-axis distance is some small positive number. Now shift the points again by the same amount in the same direction: this doubles the previous plane-axis distance. $\endgroup$ – amd Jun 14 at 21:08
  • $\begingroup$ @amd If this helps you, I have the correct answer given in the book. It is 24 cm. $\endgroup$ – Dionis Beqiraj Jun 14 at 21:23
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The only reasonable interpretation of the text is that point $B$ and $C$ lie on the circumference of the bases. Seen from "above", $BC$ is a chord of the base circle, with a length of $14\ $cm. Its distance from the center of the circle (i.e. from the axis) is thus $\sqrt{25^2-7^2}=24\ $cm.

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  • $\begingroup$ How did you calculated the length of [BC] seen from the "above" equals to 14? And what does 7 represent? Please, take a look to the picture above, and suppose that that is the position of point B and C. Thank you. $\endgroup$ – Dionis Beqiraj Jun 15 at 7:49
  • $\begingroup$ You mean 14 cm is the projection of [BC] in the bottom base? $\endgroup$ – Dionis Beqiraj Jun 15 at 8:01
  • $\begingroup$ Yes, 14 cm is the projection of [BC] on the bottom base and 7 is one half of that. But if $B$ and $C$ are not on the perimeter of the bases, as it seems to be the case in your picture, then the information given is not enough. Given the answer, I presume my interpretation is correct. $\endgroup$ – Aretino Jun 15 at 10:21
  • $\begingroup$ You are right. Actually, the given problem says that B is in the upper $circle$ and C in the bottom one. This phrase means that they are in the parameter line. One last clarification... Is there a theoreme that says that the perpendicular radius cuts a segment between 2 different perimeter points of the circle in half? $\endgroup$ – Dionis Beqiraj Jun 15 at 11:55
  • $\begingroup$ Yes: it's the theorem stating that in an isosceles triangle median and altitude relative to the base are the same. $\endgroup$ – Aretino Jun 15 at 13:38

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