Gaokao Exam Solid Geometry Proofs and Calculations 3

Question 1

A regular pyramid has a square base with side 6 and four equilateral triangular lateral faces. What is the height of the pyramid?

Accepted Answer

3√2

Answer

Wait — if lateral faces are equilateral with side 6, the slant height = 6 (from apex to base edge midpoint? No — equilateral triangle side = base edge = 6, so lateral edge = 6). Height h = √(lateral edge² - (half diagonal)²) = √(36 - 18) = √18 = 3√2.

Question 2

In a cube with edge length 2, what is the distance from the center of the cube to a vertex?

Accepted Answer

√3

Answer

The space diagonal of a cube with edge a = 2 has length 2√3. The center is halfway along the space diagonal, so the distance from center to vertex = √3.

Question 3

When setting up a coordinate system (建系) for a solid geometry problem in the Gaokao, the best choice for origin and axes is typically:

Accepted Answer

Place the origin at a vertex where three mutually perpendicular edges meet, aligning axes along those edges

Answer

In Chinese high school solid geometry, the standard approach is to set up a rectangular (Cartesian) coordinate system with origin at a right-angle vertex, with the three coordinate axes along mutually perpendicular edges. This maximizes calculation efficiency.

Question 4

In solid geometry, the 'three perpendicular lines theorem' (三垂线定理) states: If a line (OA) in a plane (α) is the projection of an oblique line (PA) (where P is outside α), and OA ⊥ l (where l is any line in α through O), then:

Accepted Answer

PA ⊥ l

Answer

The three perpendicular lines theorem: If PO ⊥ α (perpendicular from P to plane), OA is the foot's projection in α, OA ⊥ l (line in α), then PA ⊥ l. The key result is that the oblique line PA is also perpendicular to l.

Question 5

A sphere of radius R is circumscribed about a cube (the cube is inscribed in the sphere). What is the edge length of the cube in terms of R?

Accepted Answer

2R/√3 = 2R√3/3

Answer

Space diagonal of cube = 4R (diameter)? No — diagonal = 2R (diameter). Space diagonal of cube with edge a is a√3. So a√3 = 2R → a = 2R/√3 = 2R√3/3.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

Gaokao Exam Solid Geometry Proofs and Calculations 3