GATE CS 2014 Set 1 — Question 10
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General Aptitude → Quantitative Aptitude → Mensuration & Geometry
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Question
When a point inside of a tetrahedron (a solid with four triangular surfaces) is connected by straight lines to its corners, how many (new) internal planes are created with these lines? _____________
Correct answer
6 to 6
Solution
A tetrahedron has 4 vertices. Let the vertices be and the internal point be . When is connected to the corners (), the lines are formed.The new internal planes are formed by the point and pairs of vertices (which correspond to the edges of the tetrahedron). A plane is defined by 3 non-collinear points. The internal planes are the triangles formed by and the edges of the tetrahedron.The edges of the tetrahedron are . There are 6 edges in a tetrahedron.
For each edge (e.g., ), a new internal plane is created.Since there are 6 edges, there are 6 new internal planes (). These planes divide the original tetrahedron into 4 smaller tetrahedra ().Thus, 6 new internal planes are created.
For each edge (e.g., ), a new internal plane is created.Since there are 6 edges, there are 6 new internal planes (). These planes divide the original tetrahedron into 4 smaller tetrahedra ().Thus, 6 new internal planes are created.
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