FREE Gaokao Solid Geometry Proofs and Calculations Questions and Answers

Question 1

In the pyramid P-ABCD, the base ABCD is a rectangle. E is the midpoint of the side PD. Let plane α be the plane passing through points A, E, and C. What is the relationship between line PB and plane α?

Accepted Answer

PB is parallel to plane α

Answer

Let O be the intersection of the diagonals AC and BD of the rectangle. Connect EO. In triangle PBD, E is the midpoint of PD and O is the midpoint of BD, so EO is the midsegment. Therefore, EO is parallel to PB. Since EO lies within plane AEC (plane α) and PB does not, PB is parallel to plane α.

Question 2

In a cube ABCDA'B'C'D' with side length 'a', what is the sine of the angle between the line A'B and the plane BDD'B'?

Accepted Answer

1/2

Answer

Set up a coordinate system with D at the origin (0,0,0). Then A'=(a,0,a), B=(a,a,0). The vector for line A'B is v = B - A' = (0, a, -a). The plane BDD'B' is the plane x=y, so its normal vector is n = (1, -1, 0). The sine of the angle θ between a line and a plane is given by sin(θ) = |v · n| / (|v| * |n|) = |(0)(1) + (a)(-1) + (-a)(0)| / (sqrt(a²+a²) * sqrt(1²+1²)) = a / (a√2 * √2) = a / (2a) = 1/2.

Question 3

In a triangular pyramid P-ABC, the base ABC is an equilateral triangle with a side length of 2. If PA is perpendicular to the base plane ABC and PA = 3, what is the volume of the pyramid P-ABC?

Accepted Answer

√3

Answer

The volume of a pyramid is V = (1/3) * Base Area * Height. The height is given as PA = 3. The area of the equilateral triangle base is (√3/4) * side² = (√3/4) * 2² = √3. Therefore, the volume V = (1/3) * √3 * 3 = √3.

Question 4

In a regular tetrahedron with an edge length of 3, what is the distance from any vertex to the opposite face?

Accepted Answer

√6

Answer

The distance from a vertex to the opposite face is the height (H) of the regular tetrahedron. The formula for the height is H = a√(2/3) = a√6/3, where 'a' is the edge length. For a=3, the height is H = 3√6/3 = √6.

Question 5

In the right triangular prism ABC-A₁B₁C₁, ∠BCA = 90°, and AC = BC = CC₁ = 1. What is the cosine of the dihedral angle B₁-AC₁-B?

Accepted Answer

2√2/3

Answer

Set up a coordinate system with C at (0,0,0). Then A=(1,0,0), B=(0,1,0), C₁=(0,0,1), B₁=(0,1,1). The two planes are Plane(B,A,C₁) and Plane(B₁,A,C₁). A normal vector to Plane(B,A,C₁) is n₁ = vector(AC₁) × vector(AB) = (1,0,1) × (-1,1,0) = (-1,-1,1). A normal vector to Plane(B₁,A,C₁) is n₂ = vector(AC₁) × vector(AB₁) = (1,0,1) × (-1,1,1) = (-1,-2,1). The cosine of the angle θ between the planes is |n₁·n₂| / (|n₁||n₂|) = |1+2+1| / (√3 * √6) = 4/√18 = 4/(3√2) = 2√2/3.

Gaokao Exam Practice Test

Gaokao Exam Practice Test

FREE Gaokao Solid Geometry Proofs and Calculations Questions and Answers