Thursday, June 25, 2015

Unizor - Geometry3D - Lines and Planes - Problems 6

Unizor - Creative Minds through Art of Mathematics - Math4Teens

1. Prove that the distance from any point A on plane γ to plane δ, that is parallel to plane γ, is the same for any point A.
γ ∥ δ;
A∈γ; B∈δ; AB⊥δ;
A'∈γ; B'∈δ; A'B'⊥δ;
⇒ AB = A'B'

2. Given two parallel planes γ and δ and point A on plane γ.
Prove that any line parallel to plane δ passing through point A lies completely within plane γ.
γ ∥ δ; A∈γ; a ∥ δ; A∈ a
⇒ a∈γ

3. Given two intersecting planes γ and δ. Also given are line a on plane γ and line b on plane δ. These two lines, a and b are parallel to each other.
Prove that these lines are also parallel to a line of intersection c of planes γ and δ.
a∈γ; b∈δ; a ∥ b; c = γ∩δ
⇒ a ∥ c

4. Given a plane γ, line a on it and line b outside it that is parallel to line a.
Prove that any plane δ, that contains line b and not parallel to plane γ, intersects plane γ along line c parallel or coinciding with line a.
a∈γ; b∉γ; a ∥ b;
b∈δ; δ∩γ=c
⇒ a ∥ c

5. Given two skew lines a and b. Plane γ contains line a and is parallel to line b. Plane δ contains line b and is parallel to line a.
Prove that planes γ and δ are parallel to each other.
a∩b=∅; a ∦ b;
a∈γ; b ∥ γ; b∈δ; a ∥δ;
⇒ γ ∥ δ

6. Given plane γ and line a both perpendicular to line b.
Prove that these plane and line are parallel to each other.
b⊥γ; b⊥a;
⇒ a ∥ γ

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