summary

space

Welcome To Our Massive World Of Space !!!

Refresh Your Memory:


  • We need 3 NON-COLLINEAR points to form a plane.
  • Co-planer: belong to the same plane (can be intersecting-parallel or confoumded lines.)
  • planes may intersect at a line or a point.
  • Cavaliers Perspective : (way to draw 3D on a paper (the non-frontal faces ))
  1. Right angles aren't shown right - equal sides aren't shown equal (but assumed so when solving).
  2. Parallelism and Midpoints are preserved.
  3. Faces that are hidden are drawn dotted
  4. EVERYTHING is preserved in a frontal face. (ex: a rectangle is shown as a rectangle not as a parallelogram.)






Parallelism and planes :

Parallel lines and planes:
  1. Every line parallel to a line contained in a plane is parallel to this plane or contained in it.
  2. If a line (d) is parallel a plane (P), every plane passing through (d)and cutting (P), cuts it along a line parallel to (d).
  3. Every line parallel to 2 intersecting planes is parallel to their line of intersection.
  4. If 2 intersecting planes pass through 2 parallel straight lines then their line of intersection is parallel to these to lines.

Parallel Planes:

  • If two planes are parallel, every line in one of them is parallel to the other plane.
  • If a plane (P) contains two intersecting lines parallel to another plane (Q) or parallel two another 2 intersecting lines in (Q) , then (P) is parallel to (Q).
  • (P) // (Q) and (P) // (R) then, (Q) // (R).
  • If (P) // (Q) and (Q) perpendicular to (R), then (Q) perpendicular to (R).
  • If 2 planes are parallel every plane which cuts one of them cuts the other and their intersections are parallel.

If a straight line (d)//(P) DOESN'T MEAN (d)//(d') any line in (P)

Perpendicular and Planes:

BUT WE CAN'T GO BACKWARDS !!

If any straight line (D) perpendicular to any plane (P) THIS MEAN'S that (d) perpendicular to any line in (p)

common perpendicular

Is when 2 perpendicular lines have another 3rd one perpendicular to both (3 skew perp. lines)

Ex: -(AD) and (AB) =(AE)

-(GC) and (DC) = (BC)


Angle between 2 PLANES : (dihedral angle)

Steps to indicate the angle between (R) and (Q):


  1. find a common line (MN)
  2. select a st. line in (Q) perpendicular to (MN) = (AO)
  3. select a st. line in (R) perpendicular to (MN) = (OB)
THEN, the angle between (R) and (Q) is AOB


Angle between a plane and a line:

Draw an orthogonal line (CF) from the line (CH) to the plane (FGHE),

THEN FHC is the angle between (CH) and (FGHE)

Mediator plane:

DEFINITION: It is a plane perpendicular to a line at its midpoint.

Property: Any point on the M.P is equidistant from the extremities of the segment .

To Prove :

  1. By Definition.
  2. By proving 3 non-collinear points of M.P equidistant from the extremities of the segment.

Angular Bisector plane :

Definition: A plane who bisects an angle between 2 planes/lines.

Properties: Any pt. M on the plane is equidistant from the 2 sides of the angle.

How to prove:

  1. prove that angle between (P) and (R) = angle between (R) and (Q)
  2. prove that any point on (R) is equidistant from (p) and (Q) ===> then the bisector plane would be this point and the intersection line of the 2 planes.


Axis of the circle circumscribed:

Definition: It is the axis of the circle/plane where it is perpendicular to it at is center.

Property: Any point on the axis is equidistant from any pt on the circle/plane.

How to prove:

  1. By definition
  2. A pt. on the axis equidistant to 3 pts on the circle/plane + perpendicular at its center.


Some important rules:

  • Cos Rule: a^2=b^2+c^2-2bc.cosA
  • Sin Rule: a/sinA = b/sinB= c/sinC
  • Area Rule: A=1/2.absinC
  • Locus is the place where a variable point moves.. In space it is mainly a circle on plane (P) determined in different ways.
  • All Previous Geometric Rules are VERY Important!!!


THANK YOU !!

Done by: Nisrine Zaatary 11'B'