Сагс хоосон байна
Planes
A plane can be compared to the surface of water, a table or a board.
In geometry, planes are considered infinite.
In geometry, planes are considered infinite.
\latex{ S }
The relative position of points and planes
\latex{ S }
\latex{ P }
Point \latex{ P } is in plane \latex{ S }.
The distance between \latex{ P } and \latex{ S } is \latex{ 0 }.
The distance between \latex{ P } and \latex{ S } is \latex{ 0 }.
\latex{ S }
\latex{ R }
\latex{ d }
Point \latex{ R } is not in plane \latex{ S }.
The distance between \latex{ R } and \latex{ S } is \latex{ d }.
The distance between \latex{ R } and \latex{ S } is \latex{ d }.
The relative position of lines and planes
1. The line is parallel to the plane.
\latex{ S }
\latex{ R }
\latex{ e }
\latex{ d }
\latex{ e }
\latex{ S }
\latex{ \left|\right| }
They have no common points.
The distance between \latex{ e } and \latex{ S } is \latex{ d }.
The distance between \latex{ e } and \latex{ S } is \latex{ d }.
\latex{ S }
\latex{ f }
\latex{ f }
\latex{ S }
\latex{ \left|\right| }
They have an infinite number of common points.
The line is in the plane.
The distance between \latex{ f } and \latex{ S } is \latex{ 0 }.
The line is in the plane.
The distance between \latex{ f } and \latex{ S } is \latex{ 0 }.
2. The line is not parallel to the plane. The line and the plane have exactly one common point.
This is the point of intersection.
The distance between \latex{ f } and \latex{ S } is \latex{ 0 }.
This is the point of intersection.
The distance between \latex{ f } and \latex{ S } is \latex{ 0 }.
\latex{ S }
\latex{ D }
\latex{ f }
The relative position of two planes
\latex{ S_{2} }
\latex{ e }
\latex{ S_{1} }
Intersecting planes:
they have one common line. The distance between \latex{ S_{1} } and \latex{ S_{2} } is \latex{ 0 }.
they have one common line. The distance between \latex{ S_{1} } and \latex{ S_{2} } is \latex{ 0 }.
\latex{ d }
\latex{ S_{2} }
\latex{ R }
\latex{ S_{1} }
Parallel planes:
they have no common points. The distance between \latex{ S_{1} } and \latex{ S_{2} } is \latex{ d }.
Half-plane
A given line in the plane divides the plane into two half-planes. The line is part of both half-planes (unless stated differently).
\latex{ e }
half-plane
half-plane
Exercises
{{exercise_number}}. Find faces on the cube in the image that are
a) in the same plane as line \latex{ e };
b) not in the same plane as line \latex{ e };
c) parallel to line \latex{ e }. (→)
b) not in the same plane as line \latex{ e };
c) parallel to line \latex{ e }. (→)
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{\textcolor{#ef0f58}{e}}
\latex{ H }
{{exercise_number}}. Determine the common parts of the planes bordering the cube and the face \latex{ ABCD }. (→)
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{\textcolor{#ef0f58}{e}}
\latex{ H }
{{exercise_number}}. Find faces and edges parallel to the coloured faces.
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{ H }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ E }
\latex{ F }
\latex{ G }
\latex{ H }
\latex{ I }
\latex{ J }
\latex{ K }
\latex{ L }
\latex{ 1. }
\latex{ 2. }
\latex{ 3. }
{{exercise_number}}. Into how many parts can a plane be divided by
a) \latex{ 1 } line;
b) \latex{ 2 } lines;
c) \latex{ 3 } lines;
*d) \latex{ 4 } lines?
Quiz
Divide the face of a clock into three parts, so that the sum of the numbers in each section is the same.
