MATH SUMMARY

Orthogonality in space

Project by Rayan Hamdan

Hariri Highschool II

Grade 11 Sc (B)

*Figures are done using Google SketchUp 8

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)
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In the cube ABCDEFGH below, [AC] is perp. to [CD], and at the same time perp. (orthogonal) to [GH] and [EF] even when they don't intersect.
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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)
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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')

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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')

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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


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Propert 4

If H is the orthogonal projection of O on (P) such that O is any point on (DELTA), and A and B are equidistant from O, then the segments [OA] and [OB] issued from O are equal
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Property 5

(P) intersects (Q) = (d)


  • (D) belongs to (P)
  • (D') belongs to (Q)

===> (D) // (D')


Then, (d) // (D) // (D')


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Mediator Plane of a Segment

Definition


It is the plane perpendicular to [AB] at its midpoint I

Property

Any point M of the m.p. is equidistant from the extremities A and B of [AB]


MA=MB

Proof

To prove m.p., prove 3 non-collinear points on the m.p. equidistant from the extremities of the segment
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Angle Between a Line and a Plane

(d) is any non-perpendicular line passing through (P) cutting it at I

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"

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Dihedral

Definition

It is the angle between two planes


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Find the Dihedral Angle

  1. Find a common line (xy)
  2. Select a straight line in (P) perpendicular to (xy)
  3. Select a straight line in (Q) perpendicular to (xy)
  4. The dihedral angle is the one formed by the 2 perpendiculars
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Perpendicular Planes

Property 1

  1. Two planes (Q) and (P) are perpendicular if and only if one of them contains a straight line perpendicular to the other.
  2. 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.
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Property 2

If two intersecting planes are perpendicular to a third, their line of intersection is perpendicular to the third plane
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Property 3

If two planes are parallel, then every plane perpendicular to one is also perpendicular to the other.
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Common Perpendicular

(D') in (Q) an (D') // (delta)

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')

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Axis of Circle or other

Definition

(delta) is the axis of a circle since it is perp. to (C) at O

Property

Any point on (delta) is equidistant from any point on (C)

Proof

To prove axis:

1- Prove line (AO) perp. to circle (or other)

2- Prove AB=AC=AD (HA=HB=HC)

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Bisector of Dihedral

Definition

(B) is the bisector plane of (P) and (Q) since angles NIM=N'IM

[IM] is also the bisector of the dihedral

Property

The bisector (B) of the dihedral is the set of points equidistant from (P) and (Q)

M is a point on the bisector, therefore, MN=MN'

  • Any point on the bisector of the dihedral is equidistant from the planes

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THE END

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EVERYTHING IS GEOMETRY ;)