In this unit, using coordinates we will prove that segments in them are perpendicular or parallel. By this proof, we will get to know the importance and validity of the coordinate geometry. Furthermore, we will deal with the Midpoint connector theorem.

Day 4. Proof with coordinates

The proofs in this day are of three types: 1) prove lines are parallel; 2) prove lines are perpendicular; and 3) Use the previous facts to prove that a polygon is a special type of polygon.

To prove that lines are parallel or perpendicular, students first calculate their slope. They then apply either th e Parallel Lines and Slopes Theorem ( if slpoes are equal, the lines are parallel) or the Perpendicular Lines and SLopes Theorem. These steps can be repeated for as many pairs of lines are needed.

To prove that a figure is a right triangle, parallelogram, or rectangle, students simply add a third step whose justification is the definition of the figure.

Proof of Parallelgoram

Consider quadrilateral ABCD with vertices A=(0, 0), B=(8, 0), C=(11, 12), and D=(3, 12). Prove that ABCD is a parallelogram.

Solution First draw a picture, as done at the left. In the drawing, it appears that ABCD is a parallelogram. The idea is to use slopes to prove opposite sides parallel. Here is what you might write.

Using the slope formula, AD and BC have slope 4 and DC and AB have slope 0. So because of the Parallel Lines and Slopes Theorem, AD//BC and DC//AB. Thus, by the definition of Parallelogram, ABCD is a parallelogram.

Proof of Right Triangle

If T=(3, 5), O=(-1, -2), and W=(-3, 1), prove that triangle TOW is a right triangle.

Solution A drawing shows <W to be the possible right triangle. So try to show WO is perpendicular to WT. This can be done by using slope.

Slope of WO is (-2-1)/(-1-(-3))=-3/2 and slope of WT is (5-1)/(3-(-3))=4/6=2/3. So By the theorem of perpendicular lines and slopes theorem, Wo is perpendicular to slope WT. Therefore triangle TOW is a right triangle.

Day 5. The Midpoint Connector Theorem

This lesson covers three major ideas. The first is the proof of a theorem using coordinates, the midpoint connector theorem. The second idea is the notion of convenient locations for figures. The third major idea is the use of convenient coordinates that arise from a convenient location, as in the proof of the midpoint connector theorem.

A basic property of midpoint is pictured at the above. Segment MN joins the midpoints of FG and GH. It is parallel to FH. It is also half the length of FH.

These two results can be proved for any triangle. Given any triangle PQR, a coordinate plane can be put on th plane of triangle PQR so that Q is the origin and QR is on th x-axis.

Since R is on the x-axis, its second coordinate is 0. Its first coordinate is not known. We could call it x or any other unknown.Pick 2a to more easily calculate midpoints. R= (2a, 0). Neither coordinate of P is known or determined by R or Q. So call the coordinates 2b and 2c, again to more easily calculate midpoints. P=(2b, 2c). We can use these coordinates to prove the following theorem. This can be the homework of this day.

In proving a theorem,to locate the coordinate plane conveniently is very important. Convinient locations use the x-axis or y-axis as symmetry lines, or place one vertex at the origin, or both. Here are convinient locations for some other figures.

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