* 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'.
Return to Instructional Unit