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A course on the relative position of lines and planes as well as the calculation of the volume of a solid in space in the second year of high school (2de) is always a good idea. In this chapter, the student will have to know how to represent a solid in space in cavalier perspective but also to calculate its volume. Then determine the relative position between a line and a plane or between two planes.
I. Bridging perspective
In a cavalier perspective representation of a solid :
- a figure represented in a plane seen from the front is represented in true size, without changing its shape;
- two parallel lines are represented by two parallel lines ;
- aligned points are represented by aligned points ;
- the middle of a segment is represented by the middle of the drawn segment;
- the visible elements are in solid lines, those which are hidden are in dotted lines;
- a line perpendicular to the frontal plane is represented by a line making an acute angle with the horizontal of the representation support;
- any length on such a line is multiplied by a coefficient less than 1.
II. Relative positions of lines and planes
1.Rules of impact
Rules:
- Through two distinct points there passes a single line;
- Through three non-aligned points A, B, C, there passes a unique plane noted (ABC) ;
- If a plane contains two points A and B, then it contains all the points of the line (AB) ;
- If (d) is a line and A is a point not on (d), there is a unique plane containing (d) and A.
2. relative positions of two lines
Ownership:
Two lines can be :
- Coplanar: they are located in the same plane (they are secant or parallel)
- Non coplanar: and in this case they have no points in common.
3. relative positions of a line and a plane
Ownership:
A line can be :
- Contained in a plane if it passes through two points of the plane;
- Secant to the plane, if it has only one point in common with this plane (see opposite);
- Parallel to the plane if it has no common point with the plane.
4.relative position of two planes
Ownership:
Two planes are either parallel, if they have no points in common, or secant and in this case their intersection is a straight line (so they have an infinity of intersection points).
Example of intersecting planes, along the line (UV).
III- Parallelism in space
1.parallelism between lines
Properties:
- If two lines are parallel to a third, then they are parallel to each other.
- If two lines are parallel then any plane that intersects one also intersects the other.
2. parallelism between two planes
Properties:
- If two planes are parallel then any plane parallel to one is also parallel to the other.
- If two secant lines(d) and(d’) of the plane (P) are parallel to two secant lines and of the plane (P’) then the two planes (P) and (P’) are parallel.
- If two planes (P) and (P’) are parallel, then any plane that intersects one also intersects the other and the intersecting lines(d) and(d’) are parallel.
Example of parallel planes determined by two pairs of intersecting lines.
2.parallelism between lines and planes
Properties:
- If two planes are parallel and if a line is parallel to the first plane then it is also parallel to the second.
- If the line(d) is parallel to the plane (P) then any plane containing(d) and intersecting (P) intersects it along a line parallel to(d). Demonstration
- If the line(d) is parallel to a line of the plane (P) then(d) is parallel to the plane (P).
- If the planes (P) and (P’) are secant along the line and if(d) is a line parallel to the two planes (P) and (P’) then the lines and(d) are parallel.
IV. Calculations in space geometry
1.orthogonality between line and plane
Ownership:
- A line is perpendicular to a plane if and only if it is orthogonal to two intersecting lines in that plane.
- If a line is perpendicular to a plane then it is orthogonal to all lines in that plane.
2.areas and volumes of classical solids
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