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Specific Objectives

By the end of the topic the learner should be able to:

 

(a) State the geometric properties of common solids;

(b) Identify projection of a line onto a plane;

(c) Identify skew lines;

(d) Calculate the length between two points in three dimensional geometry;

(e) Identify and calculate the angle between

(i) Two lines;

(ii) A line and a plane;

(ii) Two planes.

Content

(a) Geometrical properties of common solids

(b) Skew lines and projection of a line onto a plane

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(c) Length of a line in 3-dimensional geometry

(d) The angle between

i) A line and a line

ii) A line a plane

iii) A plane and a plane

iv) Angles between skewlines.

 

 

 

Introduction

Geometrical properties of common solids

  • A geometrical figure having length only is in one dimension
  • A figure having area but not volume is in two dimension
  • A figure having vertices ( points),edges(lines) and faces (plans) is in three dimension

Examples of three dimensional figures

 

Image From EcoleBooks.comRectangular Prism

A three-dimensional figure having 6 faces, 8 vertices, and 12 edges

 

Image From EcoleBooks.comTriangular Prism

A three-dimensional figure having 5 faces, 6 vertices, and 9 edges.

 

Image From EcoleBooks.com Cone

A three– dimensional figure having one face.

 

Image From EcoleBooks.comSphere

A three– dimensional figure with no straight lines or line segments

 

Image From EcoleBooks.com Cube

A three– dimensional figure that is measured by its length, height, and width.

It has 6 faces, 8 vertices, and 12 edges

 

Image From EcoleBooks.com Cylinder

A three– dimensional figure having 2 circular faces

 

Image From EcoleBooks.com

 

 

Rectangular Pyramid

A three-dimensional figure having 5 faces, 5 vertices, and 8 edges

 

 

 

 

 

Angle between a line and a plane

The angle between a line and a plane is the angle between the line and its projection on the plane

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The angle between the line L and its projection or shadow makes angle A with the plan. Hence the angle between a line and a plane is A.

Example

The angle between a line, r, and a plane, π, is the angle between r and its projection onto π, r’.

Image From EcoleBooks.comheight is 4 m

 

 

 

 

 

 

 

Example

Suppose r’ is 10 cm find the angle

Solution

To find the angle we use tan

 

 

Angle Between two planes

Any two planes are either parallel or intersect in a straight line. The angle between two planes is the angle between two lines, one on each plane, which is perpendicular to the line of intersection at the point

Image From EcoleBooks.com

Image From EcoleBooks.com

 

 

 

 

Example

The figure below PQRS is a regular tetrahedron of side 4 cm and M is the mid point of RS;

Image From EcoleBooks.com

 

 

  1. Show that PM is cm long, and that triangle PMQ is isosceles
  2. Calculate the angle between planes PSR and QRS
  3. Calculate the angle between line PQ and plane QRS

Solution

  1. Triangle PRS is equilateral. Since M,is the midpoint of RS , PM is perpendicular bisector

     

     

    cm

    = cm

    Similar triangle MQR is right angled at M

     

     

    cm

    = cm

     

  2. The required angle is triangle PMQ .Using cosine rule

     

     

     

     

     

  3. The required angle is triangle PQM

    Since triangle PMQ is isosceles with triangle PMQ =

    (109.46)

     

     

 

End of topic  

Did you understand everything?

If not ask a teacher, friends or anybody and make sure you understand before going to sleep!

 

Past KCSE Questions on the topic.

 

 

 

1.  The diagram below shows a right pyramid VABCD with V as the vertex. The base of the pyramid is rectangle ABCD, WITH ab = 4 cm and BC= 3 cm. The height of the pyramid is 6 cm.

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(a)  Calculate the

  1. Length of the projection of VA on the base
  2. Angle between the face VAB and the base

(b)  P is the mid- point of VC and Q is the mid – point of VD.

Find the angle between the planes VAB and the plane ABPQ

 

 

 

 

2.  The figure below represents a square based solid with a path marked on it.

Image From EcoleBooks.com

 

 

 

 

 

 

 Sketch and label the net of the solid.

Image From EcoleBooks.com3.  The diagram below represents a cuboid ABCDEFGH in which FG= 4.5 cm, GH = 8 cm and HC = 6 cm

 

 

 

 

 

 

 

 

Calculate:

(a)  The length of FC  

(b)  (i)  The size of the angle between the lines FC and FH

(ii)  The size of the angle between the lines AB and FH

(c) The size of the angle between the planes ABHE and the plane FGHE

 

4.  The base of a right pyramid is a square ABCD of side 2a cm. The slant edges VA, VB, VC and VD are each of length 3a cm.

 (a)  Sketch and label the pyramid

 (b)  Find the angle between a slanting edge and the base

 

5.  The triangular prism shown below has the sides AB = DC = EF = 12 cm. the ends are equilateral triangles of sides 10cm. The point N is the mid point of FC.

Image From EcoleBooks.com

 

 

 

 

 

 Find the length of:

 (a)  (i)  BN

(ii)  EN

 (b)  Find the angle between the line EB and the plane CDEF


 




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EcoleBooks | Mathematics Form 1-4 : CHAPTER FIFTY EIGHT - THREE DIMENSIONAL GEOMETRY

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