Each Math Research Project outline consists of a *Seed Question*, *Questions to Build On*, *Extensions*, and *Basic Background*. Click here to see a list of available Math Research Project outlines.

The *Seed Question *is a question that appears in a typical math curriculum. *Questions to Build On* are simple extensions, re-interpretations, and generalizations of the* Seed Question *that a student can build a simple math research paper around, with help from a mentor (be it student or teacher).

The *Extension Questions *are bigger, more challenging extensions of the *Seed Question*. If a student can make some headway into understanding these, great! If not, they are good questions to include as part of the “Where do we go from here?” section of the paper.

The *Basic Background *consists of some of the mathematical leg-work that underlies the investigation. It can be worked on simultaneously with the investigation, which gives it context

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**Project Idea: Midpoints and Beyond**

**Seed 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 segment AB into four congruent segments?

What are the coordinates of the two points that divide *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 other such points, say, not on *AB*?

Find the point on *AB* that divides *AB *into segments whose ratio is *p/q*.

**Extension Questions**

Find the midpoint of *P*(1,4,10) and *Q*(-3, 2, 4)*.*

The midpoint of M of* **AB* is equidistant to *A *and *B*. Find *all *points in the plane that are equidistant to *A *and *B.*

Find all points in space that are equidistant to *P *and *Q*.

What is the “midpoint” of three points in the plane?

**Basic Background**

Derive the distance and midpoint formulas.

Verify that *AM* = *BM* using the distance formula.

Verify that *AM* = *BM* by constructing two congruent right triangles with *AM* and *BM* as respective hypotenuses.

*Click here to see a list of available Math Research Project outlines.*