Project #20443 - Geometry

Part 1

Note: In the attached “Rectangular Box Image” the red and green lines should be thought of as “braces” to support the three-dimensional structure of the box. Imagine it as one element of a large load-bearing structure. 

A.       Derive the distance formula (d) shown below for points A = and B =

 

Note:  Look for an application of the Pythagorean theorem where the red segment AB is the hypotenuse of a right triangle.

Note:  As this is an exercise in analytic Euclidean geometry in two-dimensions, you should only be considering the bottom of the box as on the Cartesian coordinate plane. 

1.        State each step of your derivation.

2.       Provide written justification for each step of your derivation.

B.       Derive the distance formula (d) shown below for points A =  and D =

 

Note: Look for an application of the Pythagorean theorem where the green segment AD is the hypotenuse of a right triangle.

 

1.       State each step of your derivation.

2.       Provide written justification for each step of your derivation.

C.       Create a dynamic geometry version of the attached “Rectangular Box image”(preferably in Geometer’s Sketchpad) in which you do the following: 

1.        Submit the Sketchpad file (or include screenshots that show various examples of different-shaped boxes).  The submission must do the following:

·         Allow the user to modify the size of the box by dragging on the vertices

·         Show the user the general coordinate triples (e.g.,) of all 8 vertices of the box

·         Show the user the dynamic lengths of the red, blue and green segments in the illustration

Note:  This software program does not contain 3D axes, so the lengths of the colored segments will not be correct in three-dimensional space, but this is acceptable for the requirements above.

2.       Explain (suggested length of 1-2 paragraphs) how you built the dynamic illustration. 

 

 

Part 2

Introduction:

 

Vector techniques can be used to prove and derive many geometric relationships. Use the following theorem from Euclidean geometry as you complete this assessment:

 

Theorem: A triangle and its medial triangle have the same centroid.

 

Scenario:

 

The medial triangle of a triangle ABC is the triangle whose vertices are located at the midpoints of the sides AB, AC, and BC of triangle ABC. From an arbitrary point O that is not a vertex of triangle ABC, you may take it as a given fact that the location of the centroid of triangle ABC is the vector (vector OA + vector OB + vector OC)/3.

 

Task:

 

A.  Use vector techniques to prove that a triangle and its medial triangle have the same centroid, stating each step of the proof.

1.  Provide written justification for each step of your proof.

 

B.  Provide a convincing argument short of a proof (suggested length of 3–4 sentences) that the theorem is true.  

 

Part 3

Introduction:

 

Neutral, Euclidean, hyperbolic, and spherical geometries have many interesting characteristics. It is important to understand the differences and limitations inherent in each. In this task, you will explore an interesting, counterintuitive result that will illustrate the differences found in these geometric systems.

 

Task:

 

A.  Discuss differences between neutral geometry and Euclidean geometry. 

 

B.  Explain the importance of Euclid’s parallel postulate and how this was important to the development of hyperbolic and spherical geometries. 

 

Note: Euclid’s parallel postulate states the following: “For every line l and for every external point P, there exists a unique line through P that is parallel to l.”

 

C.  The sum of the angles in a triangle varies according to the geometry in which the triangle lies.  

1.  Prove by example that the statement “There exists a triangle with a sum of angles greater than 180 degrees” is true in spherical geometry. 

2.  Prove that the statement “The sum of the angles in any triangle is 180 degrees” is true in Euclidean geometry. 

 

Note: The attached “Parallel Postulate to Triangles Diagram” may prove useful in relating the parallel postulate to triangles. The only "given" in this diagram is triangle ABC (i.e., the creation of line NM is a required step in your proof). 

 

3.  Prove by contradiction (i.e., indirect proof) that the statement, “Rectangles do not exist,” is true in hyperbolic geometry.

 

Note: You may use the hyperbolic triangle angle sum theorem as a given fact in this proof. 

 

 

Subject Mathematics
Due By (Pacific Time) 01/10/2014 04:00 pm
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