*Objectives:*

* Develop coordinate proofs for the Triangle Midsegment Theorem, the diagonals of a parallelogram, and a point of reflection across the line y = x.

* Use the concepts of the coordinate proofs to solve problems on the coordinate plane.

**The Triangle Midsegment Theorem:**

A midsegment of a triangle is parallel to a side of the triangle and has a measure equal to half of the measure of that side. To prove this theorem, one must prove the following:

Let's look at an example:

Now let us see if works for all cases:

The Midsegment Theorem can be proven using other methods.

**The Diagonals of a Parallelogram:**

The diagonals of a parallelogram bisect each other. Let's look at an example.

Now look at a general case.

It can proven also be proven in paragraph form.

**Reflection across the line y = x:**

Let's look at some examples of reflecting points across y=x to see if a pattern is seen.

(X,Y)Reflected (X,Y)(4,0) (0,4) (3,-4) (-4,3) (-2,0) (0,-2) (1,2) (2,1) The assumption can br made that a point (x,y) has a reflection (y,x). Now look at the point (4,0) and its reflection (0,4):

The definition of reflection explains that the reflection of a point P, P', over a line, y=x, will result in a segment connecting P and P' at a right angle and bisecting the segment PP'.

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