March 27, 2018

The Common Core State Standards Mathematics (CCSSM) include constructions with straightedge and compass only. Specifically, CCSSM Geometry, Congruence: Make Geometric Constructions G-CO.12 states: 

"Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line."

Here you will read about and practice these constructions. You will first do the constructions by hand and will thus need a straightedge, compass, and pencil. (You will also do the constructions using GeoGebra or Desmos.) As you do the constructions, think about why they work. 

The Math Open Reference Project (Page, John D "Constructions") includes demonstrations of numerous straightedge and compass constructions, including the basic ones described above. Read Page's short Introduction to Euclidean Construction, linked here: https://www.mathopenref.com/constructions.html. Then practice the required constructions by following along as you watch a demonstration of each required construction, linked below. 

Copying a line segment: https://www.mathopenref.com/constcopysegment.html

Copying an angle: https://www.mathopenref.com/constcopyangle.html

Bisecting a segment: (see perpendicular bisector)

Bisecting an angle: https://www.mathopenref.com/constbisectangle.html

Constructing perpendicular lines, including the perpendicular bisector of a line segment: https://www.mathopenref.com/constbisectline.html

Constructing a line parallel to a given line through a point not on the line: https://www.mathopenref.com/constparallel.html

The CCSSM Geometry Standards also include C-GO.13:

"Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle."

How would you construct an equilateral triangle inscribed in a circle? Try to do it using the Desmos canvas shown below. If you need a hint, sketch what such a construction would look like and draw radii from the circle's center to each vertex of the inscribed equilateral triangle. What are the measures of the central angles? Use this information to do the construction. Do the same for a square inscribed in a circle and a hexagon inscribed in a circle. If you get stuck on the latter, watch Page's demo linked here: https://www.mathopenref.com/constinhexagon.html

For additional details on how these topics fit into the California State Standards Mathematics, see page 134 linked here: 

https://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf.