Space
تعال نتعرف على قواعد ال"Space"
Things To Learn :
Parallel & perpendicular planes
How to prove : 2 planes parallel ,2 planes perpendicular, line perpendicular to a plane...
Angles
between two planes , between a line and a plane, angle bisector
Other rules and laws
Mediator Plane ,Axis of a circle Cos , Sin & Area law
1)Perpendicular planes
1
If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
2
If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
3
If a line (AB) is perpendicular to a plane (P) , then it is perpendicular to every line contained in the plane, ex: (CE)
2)Parallel Planes
1
2 Planes (A) and (B) are parallel if 2 intersecting lines in in (A) are parallel to 2 intersecting lines in (B)
2
If a plane intersects (q) two parallel planes(l) and (r), then the intersection is two parallel lines. (AB) parallel to (CD)
3
If two planes are perpendicular to the same line, they are parallel.
"2oo3a" Note :
3) Angles
Line&Plane
To find Angle between line and plane: Find the orthogonal projection of the line on the plane
2 Planes
To Find Dihedral Angle Between 2 Planes: 1) Find the common line (a) 2) Select a strainght line (q) & (p) from each plane , perpendicular to (a)
Bisector
To prove plane(P) bisects 2 intersecting planes(C)&(D): 1) Prove angle POF between (P) & (D) = angle POE between (C) & (P) 2)OR : Prove 3 points on (P) equidistant from (C) & (D)
4)Other Rules
Mediator Plane
Mediator plane of [AB] : plane perpendicular to [AB] at its midpoint. -Any point on it is equidistant from the 2 extremitiesA&B -To prove Mediator Plane :You can prove 3 points on it equidistant from the extremities
Axis of a circle
(OR) is Axis of circle (C) at O: -Any point on (OR) is equidistant from any point on (C). -Prove : -By defenition -RI=RK=RA & (RO) perpendicular to (C) at its center O
VIP Rules
Cos rule : a^2 = b^2 + c^2 -2bc cos(A) ___Sin rule : a/sin(A) = b/sin(B) = c/sin(C) ___Area rule : A = 0.5ab x sin (C)
"2oo3a tinsa" Note:
2)In locus : A circle of diameter [AB] becomes a sphere in space .However it's intersection with a plane gives a circle.