BÀI 4: HAI MẶT PHẲNG SONG SONG

(17 câu)

1. NHẬN BIẾT (5 câu)

Câu 1: Cho hình bình hành ABCD. Từ các đỉnh A, B, C và D lần lượt kẻ các nửa đường thẳng Ax, By, Cz và Dt song song với nhau và không nằm trong mặt phẳng (ABCD). Chứng minh mặt phẳng (Ax, By) song song với mặt phẳng (Cz, Dt).

Giải:

Ta có Cz// By nên Cz // (Ax, By)

Do tứ giác ABCD là hình bình hành nên CD // AB do đó CD // (Ax, By).\

Mà Cz và CD cắt nhau và thuộc mặt phẳng (Cz, CD).

Vậy (Ax, By) // (Cz, Dt).

Câu 2: Cho hai hình bình hành ABCD và ABEF không cùng nằm trong một mặt phẳng. Chứng minh mặt phẳng (BCE) song song với mặt phẳng (ADF).

Giải:

Vì ABCD là hình bình hành nên BC // AD .

Vì ABEF là hình bình hành nên BE // AF.

Mà BC và BE cắt nhau nằm trong mặt phẳng (BCE)

.

Câu 3: Cho hình chóp , có đáy là hình bình hành tâm . Gọi  lần lượt là trung điểm của  và .

1. a) Chứng minh rằng .
2. b) Gọi là trung điểm của là một điểm trên  và cách đều . Chứng minh rằng .

Giải:

1. a) Ta có và lần lượt là trung điểm của  và  nên  là đường trung bình trong

Tương tự  là đường trung bình trong tam giác  nên .

Lại có: .

1. b) Ta có và lần lượt là trung điểm của  và  thì  là đường thẳng cách đều  và  do vậy điểm , Do  là đường trung bình của  nên .

Ta có:

Mặt khác .

Câu 4: Cho hình chóp , có đáy là hình bình hành tâm . Gọi  lần lượt là trung điểm của .

1. a) Chứng minh rằng .
2. b) Gọi lần lượt là trung điểm của . Chứng minh .

Giải:

1. a) Ta có là đường trung bình trong tam giác .
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