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Bài tập hình học không gian ( English)

CHAPTER 1. LINES AND PLANES IN SPACE §1. Angles and distances between skew lines 1.1. Given cube ABCDA1 B1 C1 D1 with side a. Find the angle and the distance between lines A1 B and AC1 . 1.2. Given cube with side 1. Find the angle and the distance between skew diagonals of two of its neighbouring faces. 1.3. Let K, L and M be the midpoints of edges AD, A1 B1 and CC1 of the cube ABCDA1 B1 C1 D1 . Prove that triangle KLM is an equilateral one and its center coincides with the center of the cube | CHAPTER 1. LINES AND PLANES IN SPACE 1. Angles and distances between skew lines 1.1. Given cube ABCDA1B1C1D1 with side a . Find the angle and the distance between lines A1B and AC1. 1.2. Given cube with side 1. Find the angle and the distance between skew diagonals of two of its neighbouring faces. 1.3. Let K L and M be the midpoints of edges AD A1B1 and CC1 of the cube ABCDA1B1C1D1. Prove that triangle KLM is an equilateral one and its center coincides with the center of the cube. 1.4. Given cube ABCDA1B1C1D1 with side 1 let K be the midpoint of edge DD1. Find the angle and the distance between lines CK and A1D. 1.5. Edge CD of tetrahedron ABCD is perpendicular to plane ABC M is the midpoint of DB N is the midpoint of AB and point K divides edge CD in relation CK KD 1 2. Prove that line CN is equidistant from lines AM and BK. 1.6. Find the distance between two skew medians of the faces of a regular tetrahedron with edge 1. Investigate all the possible positions of medians. 2. Angles between lines and planes 1.7. A plane is given by equation ax by cz d 0. Prove that vector a b c is perpendicular to this plane. 1.8. Find the cosine of the angle between vectors with coordinates ai bi ci and a2 b2 c2 . 1.9. In rectangular parallelepiped ABCDA1B1C1D1 the lengths of edges are known AB a AD b AAi c. a Find the angle between planes BB1D and ABC1. b Find the angle between planes ABi Di and Ai Ci D. c Find the angle between line BDi and plane AiBD. 1.10. The base of a regular triangular prism is triangle ABC with side a. On the lateral edges points Ai Bi and Ci are taken so that the distances from them to the plane of the base are equal to 2 a a and a respectively. Find the angle between planes ABC and A1B1C1. Typeset by AmS-TEX 1 2 CHAPTER 1. LINES AND PLANES IN SPACE 3. Lines forming equal angles with lines and with planes 1.11. Line l constitutes equal angles with two intersecting lines 11 and 12 and is not perpendicular to plane n that contains these lines. Prove that .