MATH SUMMARY
Orthogonality in space
Project by Rayan Hamdan
Hariri Highschool II
Grade 11 Sc (B)
*Figures are done using Google SketchUp 8
Email: rayanh_96@hotmail.com
Location: Beirut, Lebanon
Orthogonal & Perpendicular Lines
- When two lines are parallel then every line perpendicular (orthogonal) to one is also perpendicular to the other
Orthogonality of lines
- (d) intersects and is perpendicular to (d')
- (d) is also perpendicular (orthogonal) to (d'') even when (d'') does not intersect with (d)
Line Perpendicular to a Plane
- A line (d) is perpendicular to a plane (P) if and only if (d) is orthogonal to two intersecting lines of (P)
Properties
Property 1
- If two lines are parallel, then every plane perpendicular to one is also perpendicular to the other
If (d) // (d') and (P) perp. (d) , then (P) perp. (d')
- If two lines are perpendicular to the same plane, they are parallel
If (d) perp. (P) and (d') perp. (P) , then (d) // (d')
Property 2
- If two planes are parallel, then every line perpendicular to one is perpendicular to the other
If (P) // (P') and (d) perp. (P) , then (d) perp. (P')
- If two planes are perpendicular to the same line, then they are parallel
If (P) perp. (d) and (P') perp. (d) , then (P) // (P')
Property 3
- If a line (d) is perpendicular to a plane (P) , then every line orthogonal to (d) is in (P) or perpendicular to (d)
If (d) perp. (P) and (d') perp. (d), then (d') belongs to (P) or (d') // (P)
- If a line is perpendicular to a plane, then every line parallel to the plane is orthogonal to this line
If (d) perp. (P) and (d') // (P), then (d') perp. (d)
- If a line is perpendicular to a plane, then every line contained in (P) is perpendicular to this line
- If a line is parallel to a plane, it is parallel to all lines in the plane
Propert 4
Property 5
- (D) belongs to (P)
- (D') belongs to (Q)
===> (D) // (D')
Then, (d) // (D) // (D')
Mediator Plane of a Segment
Definition
It is the plane perpendicular to [AB] at its midpoint I
Property
MA=MB
Proof
Angle Between a Line and a Plane
H is the orthogonal projection of A on (P)
(d') = (IH) is orthogonal projection of (d) = (IA) on (P)
The angle between the line and the plane is "a"
Dihedral
Definition
Find the Dihedral Angle
- Find a common line (xy)
- Select a straight line in (P) perpendicular to (xy)
- Select a straight line in (Q) perpendicular to (xy)
- The dihedral angle is the one formed by the 2 perpendiculars
Perpendicular Planes
Property 1
- Two planes (Q) and (P) are perpendicular if and only if one of them contains a straight line perpendicular to the other.
- If two planes (Q) and (P) are perpendicular, then every line in one of them perpendicular to their line of intersection ([AB]) is perpendicular to the other.
Property 2
Property 3
Common Perpendicular
The perp. to (delta) in (Q) cut (D') at B
(AB) is in (Q) and perp. to the inters. of (P) and (Q)
Then, (AB): perp. (P) ===> perp. (D)
So, (AB) perp. to (D) and (D')
===> [AB] is common perp. to (D) and (D')
Axis of Circle or other
Definition
Property
Proof
1- Prove line (AO) perp. to circle (or other)
2- Prove AB=AC=AD (HA=HB=HC)
Bisector of Dihedral
Definition
[IM] is also the bisector of the dihedral
Property
M is a point on the bisector, therefore, MN=MN'
- Any point on the bisector of the dihedral is equidistant from the planes