5.7 Proofs Using Coordinate Geometry

Geometry

Holt, Rinehart, and Winston

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'.