Geometry - intersections on objects?

Question

Here are three questions on intersections, all similar: thanks for your help!

1. Which does NOT represent a possible intersection of a line and triangle?
a. 0 points
b. 1 point
c. 2 points
d. 3 points

2. Which does NOT represent a possible intersection of two spheres?
a. 0 points
b. 1 point
c. 2 points
d. infinitely many points

3. Which does NOT represent a possible intersection of a line and a right cylinder?
a. 3 points
b. 2 points
c. 1 point
d. a segment

Please briefly explain your choices, thanks!

Answer 1

1. (d) three points.

A line passing through a triangle in the same plane as the triangle will touch two points (c) on its side. A line tangent to one vertex of a triangle will touch one point (b). A line that doesn't touch the triangle at all will intersect no points (a).

Note that it's clear from the possible answers to this question, that only intersecting with the 'surface' (or 'edge' in two dimensions) of a shape is counted. Otherwise, 2 points AND 3 points on a triangle would be impossible -- counting the interior the only possibilities are 0 points, 1 points, and a segment.

2. (c) two points.

Two spheres might not intersect at all (a), or touch at one point (b) if they are tangent. Or they can intersect with a circle (which counts as "infinitely many points) if they pass through each other (d).

3. (a) three points

A line could be tangent to a cylinder at just one point, whether it's the curved side or the rim (c). A line could coincide with the side or end of a cylinder in which case the intersection is a segment (d) (for example a line in the same plane as the cylinder's base, and passes through the center of the base, the line intersects the base as a segment that is a diameter on the base) . A line passing through the cylinder will intersect [its surface] in two places (b). That leaves (a) as the impossible one.