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  Construction Homework Help - K-12 Grade Level, College Level Mathematics

Introduction of Constructions:

The drawing of various shapes using only a compass and straightedge or ruler is called Construction. Measurement of lengths or angles is not allowed.

Compass and straightedge constructions

Compass and straightedge (ruler) construction are the construction of lengths, angles and other geometric figures using only an idealized ruler and compass.

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The idealized ruler, known as a straightedge is assumed to be infinite in length and has no markings on it and only one edge. Compass is assumed to collapse when lifted from the page so may not be directly used to transfer distances. (This is not an important restriction, as this may be achieved via the compass equivalence theorem.) Formally, the only permissible constructions are those granted by Euclid's first three postulates.

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Every point constructible using compass and straightedge may be constructed using compass alone.

Compass and straightedge tools

The "compass" and "straightedge" of compass and straightedge constructions are idealizations of rulers and compasses in the real world:

  • The compass can be opened arbitrarily wide, but it has no markings on it. Circles can only be drawn by using two existing points which give the centre and a point on circle. The compass collapses when not used for drawing; it cannot be used to copy a length to another place.
  • The straightedge is infinitely long but it has no markings on it and has only one edge, unlike ordinary rulers. It only used to draw a line segment between two points or to extend an existing line.

The basic constructions

  • Creating line through two existing points.
  • Creating circle through one point with centre another point
  • Creating point which is the intersection of 2 existing, non-parallel lines
  • Creating 1 or 2 points in the intersection of a line and a circle (if they intersect)
  • Creating 1 or 2 points in the intersection of two circles (if they intersect).

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