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
GATE 2024 · General Aptitude · Patterns In Three Dimensions · medium
Answer: The outer boundary of the cross-section is a rhombus, formed by the two identical isosceles triangular sections of each cone joined at their shared base diameter.
- Identify the geometry of one cone's axial cross-section: Let each cone have height h and base radius r. An axial plane (through the apex and the base center) cuts the cone along two slant edges, giving an isosceles triangle with base 2r and two equal sides l = sqrt(r^2 + h^2).
- Combine both cones' cross-sections: The upper (inverted) cone contributes an isosceles triangle pointing upward; the lower cone contributes one pointing downward. They share the base edge of length 2r. The combined figure has four sides, all equal to l = sqrt(r^2 + h^2). The vertical diagonal is 2h (top apex to bottom apex) and the horizontal diagonal is 2r (left edge of base to right edge of base). A quadrilateral with four equal sides and diagonals bisecting each other is a rhombus.
- Verify it is not an ellipse or hexagon: The cut only intersects flat lateral surfaces of the cones (ruled surfaces along straight lines), so the boundary edges are straight line segments, ruling out an ellipse. There are exactly four intersection points (top apex, bottom apex, left base edge, right base edge), so only four sides exist, ruling out a hexagon.