Visualize two identical right circular cones such that one is inverted over…
2024
Visualize two identical right circular cones such that one is inverted over the other and they share a common circular base. If a cutting plane passes through the vertices of the assembled cones, what shape does the outer boundary of the resulting cross-section make?
- A.
A rhombus
- B.
A triangle
- C.
An ellipse
- D.
A hexagon
Attempted by 2 students.
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Correct answer: A
Answer: A rhombus.
The plane passes through both vertices, so it contains the line joining the vertices (the common axis of the two cones).
A plane that contains the axis of a right circular cone intersects that cone in an isosceles triangle; the base of that triangle is a diameter of the common circular base.
Applying this to both identical cones gives two congruent isosceles triangles that share the same base diameter. Placing them together along that diameter produces a symmetric four-sided figure whose four edge lengths are equal (they are the slant edges of the cones).
Therefore the outer boundary of the cross-section is a rhombus.