# Right Circular Cylinder: Distance between axis and plan

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.

• Impossible to determine with the information at hand. These constraints don’t prevent $BC$ from intersecting the axis, for instance. – amd Jun 14 at 20:40
• @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. – Dionis Beqiraj Jun 14 at 20:55
• 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. – amd Jun 14 at 21:08
• @amd If this helps you, I have the correct answer given in the book. It is 24 cm. – Dionis Beqiraj Jun 14 at 21:23

## 1 Answer

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.

• 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. – Dionis Beqiraj Jun 15 at 7:49
• You mean 14 cm is the projection of [BC] in the bottom base? – Dionis Beqiraj Jun 15 at 8:01
• 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. – Aretino Jun 15 at 10:21
• 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? – Dionis Beqiraj Jun 15 at 11:55
• Yes: it's the theorem stating that in an isosceles triangle median and altitude relative to the base are the same. – Aretino Jun 15 at 13:38