Título del libro

BASIC GEOMETRY CONSTRUC-TIONS

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Sección 1

CONCEPTS AND DEFINITIONSBASIC GEOMETRY CONCEPTS AND DEFINITIONS

Geometry: is  a  branch  of  mathematics  concerned  with  questions  of  shape,  size,  relative  position  of  figures,  and  the  properties  of  space.  

Point: an  entity  that  has  a  location  in  space  or  on  a  plane,  but  has  no  extent.

Line: a  concept  which  includes,  but  is  not  limited  to,  an  infinitely-­‐extended  one-­‐dimensional  figure  with  no  curvature

Line segment: It´s  a  line  with  two  end  points

Ray: a  part  of  a  line  which  is  finite  in  one  direction.  It  begins  in  a  particular  point  (initial  point)

End point: it´s  the  point  where  a  line  segment  ends

Mid point: it´s  the  point  that  is  half  way  between  the  initial  point  and  the  end  point  on  a  line  segment

Length: measurement  of  a  segment

GEOMETRY DRAWING AND SUPPLIES. CONCEPTS AND DEFINITIONS,

Freehand: It  consists  in  drawing  by  hand  without  any  guiding  indtruments.  

Technical drawing: it´s  a  drawing  made  with  the  help  of  instruments  or  supplies.

Compass: is  a  tool  for  drawing  arcs  and  circumferences  and  also  for  measuring  lengths.  It  has  two  arms  linked  by  a  hinge

Protactor: It  is  an  instrument  used  to  measure  or  to  draw  angles  on  paper.

Set triangles (US) or set square (UK): they  are  two  tools  with  a  triangular  shape.  One  is  caled  the  45º  triangle  (or  the  45º  square)  and  the  other  the  30º  or  60  º  triangle  or  square.  Both  of  them  have  a  90º  angle.  Using  both  of  them  at  the  same  time  you  can  draw  parallel  and  perpendicular  lines,  and  different  angles  measumerement.

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Eraser: it  is  an  object  used  to  delete  on  the  paper  

Marker or felt-tip pen: it  is  a  pen  which  has  its  own  ink-­‐source,  and  usually  a  tip  made  of  porous,  pressed  fibers.

Ruler: it  is  an  instrumemnt  used  to  draw  straight  lines  or  to  measure  the  distance  between  two  points.

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TIP: Visit http://mathisfun.com/geometry/constructions.html

CONSTRUCTIONS

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Sección 2

CONSTRUCTIONS1. OPERATIONS WITH LINE SEGMENTS

The  purpose  of  this  section  is  to  perform  some  operations  with  line  segments.  It  is  very  important  that  you  always  use  the  compass  to  take  lenghts,  to  copy  or  to  move  them.  You  don´t  need  the  ruler  fot  anything  but  to  draw  the  rays.  

a) Line  segment  copying:  Given  the  segment  AB,  copy  it.  It  must  have  the  same  lenght.

STEP  1:  Draw  a  ray  from  point  A.  You  can  place  point  1  wherever  you  want

b) Line  segment  addition:  Given  the  line  segments  AB,  CD  and  EF,  add  them.  Eraser:  it  is  an  object  used  to  dele-­‐

te  on  the  paper  

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STEP  2:  Using  the  compass  take  the  lenght  AB2

STEP  2:  Using  the  compass,  take  AB  lenght,  adn  copy  it  on  ray  from  A´.  You  obtain  point  B´.

3 STEP  3:  Keep  that  length  in  your  compass,  set  it  on  the  beginning  of  the  ray  (point  A´)  and  draw  an  arc  obtaining  B´.

4 STEP  4:  Finally  enhance  with  black  ink  the  result  A´B´.

STEP  1:  Draw  a  ray  from  point  A´

STEP  3:  Take  CD  length  with  the  compass,  and  copy  it  from  point  B´.  You  obtain  point  C´.

STEP  4:  Repeat  step  2  or  step  3  using  EF  lenght  and  point  D´.  You  obtain  point  F´.

STEP  5:  The  result  is  the  total  length  of  the  three  segments  copied  one  after  the  other.  Remembre  to  enhance  the  result  with  black  ink.

c) Line  segment  substraction:  Given  the  line  segments  AB,  CD  substract  them.  

STEP  1:  Draw  a  ray  from  point  A´

STEP  2:  Using  the  compass,  take  AB  lenght,  adn  copy  it  on  ray  from  A´.  You  obtain  point  B´.

STEP  3:  Take  CD  length  with  the  compass,  and  copy  it  from  point  A´  (or  C´).  You  obtain  point  D´.

STEP  4:  The  result  of  AB-­‐CD  is  seg-­‐ment  D´B´.  You  must  enhance  it  with  black  ink.

2. PERPENDICULAR SEGMENT LINE BISECTOR

Draw  the  perpendicular  segment  line  bisector  when  you  know  the  segment  AB.

3. ANGLE BISECTOR

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STEP  1:  Draw  two  arcs  with  the  sa-­‐me  radius  using  points  A  an  B  (cen-­‐ter  os  the  arcs).  You  will  get  points  1  and  2  where  both  arcs  intersect

STEP  2:  Join  point  1  and  point  2  and  you  will  get  the  perpendicular  segment  line  bisector

STEP  1:  Make  an  arc  cen-­‐tered  in  point  V  with  the  radius  you  want

STEP  2:  Centered  in  the  points    1  and  2,  with  the  same  radius,  draw  two  arcs.  They  intersect  in  point  3.

STEP  3:  Join  point  3  with  point  V  and  you  will  ob-­‐tain  the  angle  bisector

2. CIRCLES

Circle: set  of  points  equidistant  to  a  point  called  center

Circumference:  total  length  of  the  circle

Center:  point  equidistant  to  any  point  on  the  circle

Radius:  distance  between  the  circunference  an  the  center

Diameter: Line  segment  that  passes  through  the  center  and  connects  two  points  of  the  circunference

Chord:  line  segment  that  joins  two  points  of  the  circunference  (  it  doesn´t  pass  through  the  center)

Tangent:  line  that  intersects  the  circumference  in  one  point

Outer circle:  circle  that  is  outside  the  circumference

HOW TO DRAW A CIRCLE KNOWN THREE POINTS (A,B,C)

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STEP  1:  Join  point  A  with  point  B.  Trace  its  perpendicular  bisector.

STEP  2:  Join  points  B  and  C  (or  A  and  C).  Trace  its  perpendicular  bisector.  Both  perpendicular  bisector  inter-­‐sect  in  the  center  of  the  circle.

STEP  3:  Draw  the  circumference  with  the  centre  in  that  point  and  radius  to  any  of  the  given  points.

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A

BC

DRAW  THE  CIRCLE  GIVEN  THREE  POINTS  OF  ITS  CIRCUMFERENCE

A

BC

DRAW  THE  CIRCLE  GIVEN  THREE  POINTS  OF  ITS  CIRCUMFERENCE

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ADD  THESE  SEGMENTS

B

SUBSTRACT  THESE  SEGMENTS

B

DIVIDE  THIS  SEGMENT  IN  5  PIECES:  you  have  to  use  THALES

AC D

AC D

C D

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MAKE  THESE  OPERATIONS  WITH   KNOWING  THESE  SEGMENTS

AC D

B E F

A)  AB + CD

4C)

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AB +34

CD −15

EFC)            12 AB + 2

4 CD − 13 EF

CD6

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CIRCLES  PATTERN

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DRAW  A  PENTAGON  WITH  SIDE=65  CM

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DRAW  A  CIRCUMFERENCE  WITH  RADIUS  R=5  CM  AND  IN  IT,  AN  HEXAGON

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DRAW  THE  STAR  SHAPE  IN  A  HEPTAGON.  THE  HEPTAGON  IS  INSIDE  A  CIRCLE  WHICH  HAS  A  5  CM  RADIUS

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DRAW  THE  STAR  SHAPE  IN  A  DECAGON.  THE  HEPTAGON  IS  INSIDE  A  CIRCLE  WHICH  HAS  A  4  CM  RADIUS

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DRAW A PENTAGON. SIDE = 6 CM

DRAWING

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DRAW A 6 CM SIDE HEXAGON

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DRAWING

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DRAW A HEPTAGON. SIDE LENGHT 5 CM

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DRAWING

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DRAW A 4 CM LENGHT SIDE OCTAGON

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DRAW A 4CM LENGHT 9 SIDES POLYGON

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DRAW A DECAGON (SIDE LENGHT=4 CM) AND DRAW ALL ITS POSSIBLE STAR SHAPES

DRAWING

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DRAW A DECAGON (10 SIDES; L=4CM) AND ITS STAR SHAPE POLYGON. GIVE IT A 6 CM WIDTH.

DRAWING

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DRAW A 11 SIDES POLYGON WITH L=4CM AND ONE OF ITS STAR SHAPES. GIVE IT A 6 CM WIDTH.

DRAWING

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DRAWING

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