GeometryResearch writing prompt

This is a mathematical research paper. You will respond to the prompts given below in paragraph form using complete, organized sentences, in 3-5 pages. You may use outside sources, but they must be cited correctly. Any plagiarism will result in a zero.

First rough draft due Friday, April 4. Final copy due Friday, April 11.First rough draft must have an introduction, answer to seed question, and at least two other questions/background topics.

Option #1: Midpoints and BeyondSeed question:

Find the midpoint M of the segment with endpoints A(3, 1) and B(15, 9).Questions to build on:

The midpoint M divides the segment AB into two congruent segments. What are the coordinates of the three points that divide the segment AB into four congruent segments?

What are the coordinates of the three points that divide the segment AB into three congruent segments?

Find the point on AB that is twice as close to A as it is to B. Are there any such points not on AB?

Extension questions: Find the midpoint N of P(1, 4, 10) and Q(-3, 2, 4). The midpoint M of AB is equidistant to A and B. Find all points in the plane

that are equidistant to A and B. What is the “midpoint” of three points in the plane?

Basic background: Derive the distance and midpoint formulas. Verify that AM = BM using the midpoint formula. Verify that AM = BM by constructing two congruent right triangles with AM

and BM as respective hypotenuses.

Option #2: Pythagorean TriplesSeed question:

Find some triples of positive numbers that satisfy the equation a2+b2=c2.Questions to build on:

Find all triples of positive numbers with c<100that satisfy the equation a2+b2=c2.

Show that a multiple of a Pythagorean triple is also a Pythagorean triple. Find all Pythagorean triples whose legs are consecutive integers. Find all Pythagorean triples whose legs differ by two

Find all Pythagorean triples that consist of a pair of consecutive integers.Extension questions:

Show that a primitive Pythagorean triple always consists of at least one multiple of three

Find some primitive Pythagorean quadruples. Find some integer solutions to the equation a2+b2=c3.

Basic background: Verify a couple of Pythagorean triples. Prove the Pythagorean theorem.

Option #3: Platonic SolidsSeed question:

What are the 5 platonic solids? Where does each solid get its name?

Questions to build on: What is a net of a solid figure? Give descriptions of each of the solids, including the number of faces, edges,

and vertices and the shapes of the faces. What properties of a solid is require for it to be considered “Platonic?”

Extension questions: Build the five platonic solids with paper. What are the regular 2-dimensional figures that can be tessellated in the

plane? How are they related to the Platonic solids? Create a tessellation using all those figures.

Basic background: With what element of the cosmos did Plato associate each solid?