Geometric Constructions © T Madas

Midpoints & Perpendicular Bisectors Constructing Midpoints & Perpendicular Bisectors © T Madas

Constructing the Midpoint & Perpendicular bisector of a line segment Construction Lines C A B Construction Lines D © T Madas

Constructing Angle Bisectors © T Madas

Constructing an angle bisector © T Madas

Equilateral Triangles Constructing Equilateral Triangles © T Madas

Constructing an equilateral triangle © T Madas

Constructing a perpendicular to a given point on a line © T Madas

Constructing a perpendicular to a given point on a line segment Construct the perpendicular bisector of this segment © T Madas

Constructing a perpendicular to a given point on a line segment Alternative Construction Why does it work? © T Madas

Constructing a perpendicular to a given point outside a line © T Madas

Constructing a perpendicular from a given point outside a line segment Construct the perpendicular bisector of this segment © T Madas

an identical triangle to one given Constructing an identical triangle to one given © T Madas

Constructing an identical triangle to one given © T Madas

an identical angle to one given Constructing an identical angle to one given © T Madas

Constructing an identical angle and sides to one given © T Madas

Constructing an identical angle to one given © T Madas

a parallel line to a given line, Constructing a parallel line to a given line, through a given point © T Madas

Constructing a parallel line to a given line, through a given point Why does it work? © T Madas

Constructing a parallel line to a given line, through a given point Alternative Construction Why does it work? © T Madas

Circumscribing a triangle © T Madas

Circumscribing a triangle The 3 perpendicular bisectors of the sides of a triangle are concurrent. The point they meet is called the Circumcentre. The Circumcentre has the property of being equidistant from all three vertices of the triangle. For this construction: We need the intersection of two perpendicular bisectors (since all three all concurrent). © T Madas

Inscribing a triangle in a circle © T Madas

Inscribing a triangle in a circle The 3 angle bisectors of a triangle are concurrent. The point they meet is called the Incentre. The Incentre has the property of being equidistant from all three sides of the triangle. For this construction: We need the intersection of two angle bisectors (since all three all concurrent). © T Madas

Dividing a line into a given number of equal segments © T Madas

Dividing a line into a given number of equal segments Suppose we want to divide AB into 3 equal segments A B © T Madas

The midpoint of a line segment Construct: The midpoint of a line segment The perpendicular bisector of a line segment The angle bisector of an acute angle The angle bisector of an obtuse angle An equilateral triangle A triangle with sides: 3 cm, 4 cm and 6 cm A 45° angle A 60° angle A 120° angle The perpendicular to a line segment through a given point on the line segment The perpendicular to a line segment through a given point outside the line segment A parallel line to a line segment which passes through a point outside the line segment © T Madas

Constructing Regular Hexagons © T Madas

Constructing a Regular Hexagon Why does it work? © T Madas

Constructing a Regular Hexagon Why does it work? © T Madas

Constructing a Regular Hexagon Why does it work? © T Madas

Constructing Regular Pentagons © T Madas

Constructing a Regular Pentagon Start with the circumscribing circle Draw a diameter Draw the perpendicular bisector of that diameter Mark its intersection with the circle Bisect the radius as shown Draw arc as shown and mark its intersection with the diameter This is the required side length for a regular pentagon, which is circumscribed by the circle © T Madas

Constructing Regular Octagons © T Madas

Constructing a Regular Octagon © T Madas

Constructing Regular Decagons © T Madas

Constructing a Regular Decagon To construct a decagon you need a regular pentagon first Bisect one of the sides of the pentagon Use the chord of the new arc produced to construct a regular decagon You can use this idea to constuct a dodecagon from a hexagon A 16 – sided regular polygon using an octagon An eicosagon from a decagon and so on © T Madas

© T Madas