## Tuesday, May 12, 2015

### Unizor - Geometry3D - Lines and Planes - Problems 1

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Problem 1
Given a straight line a in three-dimensional space and a point M outside of it.
Construct a line b that passes through a given point M parallel to a given line a.

Solution
There is one and only one plane μ that can be constructed by a given line a and a point M outside it. After constructing this plane μ=μ(M,a), we should construct a line b within this plane μ that passes through a given point M and parallel to a given line a. It's a plane geometry construction that we, supposedly, are familiar with.
That is the line we need.

Problem 2
Given a plane μ and a point M outside of it.
Construct a plane ν that passes through a given point M parallel to a given plane μ.

Solution
On a given plane μ construct two intersecting straight lines a and b.
Construct two corresponding lines a' and b' that pass through a given point M parallel to two lines a and bon the plane μ.
These two new lines a' and b' determine one and only one plane ν that is parallel to a given plane μ, according to a theorem proven in a lecture about parallel planes.

Problem 3
Given two non-intersecting non-parallel lines (skew lines) a and b in three-dimensional space.
Construct a plane μ that passes through line a parallel to line b.

Solution
Choose a point M on line a and construct a new line b' that passes through this point M parallel to line b.
The two intersecting lines a and b' determine one and only one plane μ that is parallel to line b', according to a theorem proven in a lecture about parallel lines and planes.

Problem 4
Given two non-intersecting non-parallel lines (skew lines) a and b in three-dimensional space and a point M not lying on any of them.
Construct a line c that passes through a given point M and intersects both line a and line b.

Solution
The line that connects point M and both skew lines a and b must lie on the plane that is defined by point M and line a as well as on the plane defined by the same point M and another line b.
Therefore, it's the line of intersection of these two planes, each of which can easily be constructed by a point and a line it should contain.