Welcome To Our Massive World Of Space !!!
- 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 ))
- Right angles aren't shown right - equal sides aren't shown equal (but assumed so when solving).
- Parallelism and Midpoints are preserved.
- Faces that are hidden are drawn dotted
- EVERYTHING is preserved in a frontal face. (ex: a rectangle is shown as a rectangle not as a parallelogram.)
Parallelism and planes :
- Every line parallel to a line contained in a plane is parallel to this plane or contained in it.
- 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).
- Every line parallel to 2 intersecting planes is parallel to their line of intersection.
- If 2 intersecting planes pass through 2 parallel straight lines then their line of intersection is parallel to these to lines.
- 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:
If any straight line (D) perpendicular to any plane (P) THIS MEAN'S that (d) perpendicular to any line in (p)
Angle between a plane and a line:
THEN FHC is the angle between (CH) and (FGHE)
Property: Any point on the M.P is equidistant from the extremities of the segment .
To Prove :
- By Definition.
- By proving 3 non-collinear points of M.P equidistant from the extremities of the segment.
Angular Bisector plane :
Properties: Any pt. M on the plane is equidistant from the 2 sides of the angle.
How to prove:
- prove that angle between (P) and (R) = angle between (R) and (Q)
- 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:
Property: Any point on the axis is equidistant from any pt on the circle/plane.
How to prove:
- By definition
- 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!!!