Five Proofs
Nicole Mosteller EMAT 6690

The NCTM Standards 2000 state that "an emphasis on mathematical connections helps students recognize how ideas in different areas are related." One of the most important ideas that I try to impress upon my students is that every aspect of Mathematics can be related in some way. Many students spend an entire semester/year working on the "small pieces" of a subject without ever seeing the big picturre. One such mathematics course where topics need more connection is the typical high school Geometry class. This instructional unit is designed to show how one idea can be connected throughout a Geometry course.

Granted, technology is not needed for these proofs to be completed. However, I strongly urge the use of some Geometry technology (i.e., Geometer's Sketch Pad). Many students get a better grasp of proof when given a clear visual aid which also allows for easy manipulation. Technology allows students the ability to make the conjecture "Will this be true for every such case?".

The Lesson Plan

Included with this set of proofs is a proposed course guideline. These proofs are not be completed all in one day or even within one unit, but to be used as supplemental material as the high school Geometry course progresses. See Lesson Plan.

The Proposition

Show that the following can be proven using five different concepts discussed during the typical high school Geometry course.

Given: Right triangle XYZ with the
midpoint of the hypotenuse labeled M.
Prove: The midpoint of the hypotenuse of a right triangle
is equidistant from all vertices of the triangle.

The Proofs

Proof #1: This proof requires application of the definitions in relation to the parts of a circle and properties of inscribed angles.

Proof #2: This proof requires the application of properties of parallelograms and rectangles.

Proof #3: This proof requires the application of parallel line properties and isosceles triangle properties.

Proof #4: This proof requires applications of reflection of a triangle over a line and similar triangles.

Proof #5: This proof involves the application of the Midpoint Formula and the Distance Formula both in relation to Coordinate Geometry.

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