Study Suggestions for Quiz 2

The quiz will emphasize topics that have been central in the time since the midterm, but of course everything depends on earlier theorems and constructions, so there is no wall between recent topics and earlier topics.

Ratios and Similar Triangles

The concept of ratio, proportionality and similarity is one of the biggest in geometry. SIGNED RATIO became very inportant in recent weeks.

Ratios show up in

• Similar Triangles
• Dilations
• Transversals of Parallels
• How an angle bisector in a triangle cuts the opposite sides

And ratio is applied to

• Coordinates
• Scaling relationships of area and volume
• Midpoint figures
• Centers of mass
• Proportions in trapezoids
• Common tangents of circles

Areas and Volumes

Area

• Formulas for areas of triangles, trapezoids, parallelograms
• Effect of scaling on area
• Effect of shearing on area. Cavalieri principle.

Volume

• Formula for volume of a cube and rectangular solid
• Effect of scaling on volume.
• Formula for volume of pyramids and cones based on "1/3" pyramid in cube. This is based on effect of shearing on area. Cavalieri principle.
• Relations among volumes of Platonic solids that are nested within one another.

Polyhedra

• Definition of a regular polyhedron
• Names and numbers of vertices, edges, and faces of Platonic solids (and how many edges and faces at a vertex).
• How Platonic solids fit into one another. Application of scaling of volume.
• Computing dihedral angles for examples of polyhedra.
• Duals of polyhedra.
• Informal explanation of what is an archimedian polyhedron. What is truncation?

Barycentric and affine coordinates

• How to get from ratios of lengths to coordinates, as shown in many problems and example.
• The parallels and cevian lines in the figure with a point P in a triangle, the paralles to sides through it, etc., as in assignments.
• How to find coordinates of a point.
• How to construct a point from coordinates.
• How to read off coordinates in a lattice.
• Ceva and Menelaus theorems.