Math 445 Lab 1, Part A

Affine Coordinates in the Plane

Making affine coordinates in the plane

In a new sketch, draw 3 points ABC and a point P.  Then we will make a tool that computes a ratio that is one of the affine coordinates. 

• Construct the LINES (not segments) AB, BC, CA.
• Construct a line through P parallel to line AB.  Let P' be the intersection of this line with line AC.
• Select points A, C, P' in order and then Measure>Ratio to get the ratio AP'/AC.

We will denote this ratio as h_C(P) or as h[C](P); we will call it the affine C-height of P above base AB.

Make a Tool

Hide the lines.  Then make a tool that will take A, B, C, P and compute h_C(P).

• Check the order of the points using script view.  You can reorder the points by dragging if necessary.
• For a tool that makes great labels, go into script view.  Double-click on the measure m1. Then in the dialog, click on the box "Use label in Sketches" and make the label of the ratio the following (exactly - this means that the first character in the name is the equal sign = and then be sure that the brackets around 3 and 4 are braces, also called curly brackets):
  =h[{3}]({4})

Exploring

Question: For what points P is c(P) equal the following values:  0, 1, 1/2, -1, 2, 1/3?

• Construct triangles ABC and ABP and measure the ratio of areaABP/area ABC. 
• How does this ratio relate to c(P)?

Construct a new line parallel to AB and Merge point P to this line. 

• As P moves along the line, how does h_C(P) change? 
• How can this be explained in terms of the definition of h_C(P), as a ratio?
• How can this be explained in terms of the area ratio?

Two Coordinates

Continue with triangle ABC and point P, but now use your tool to compute both h[C](P) and h[B](P).  You can use x and y to denote y = h[C](P) and x = h[B](P). 

Construct both triangles ABP and CAP and measure the areas and the ratios obtained by dividing the areas by the area of ABC.  These ratios should agree (except possible for sign) with x and y.

Equation of a line

Draw a line AD and merge point P with this line.  Check that as P moves along the line, the ratio y/x is constant.  Check that the constant equals h[B](D)/h[C](D). 

Use this and the calculator to find quantities a and b so that ax + by = 0 for all P on this line.  (So there should be a quantity ax + by calculated on the screen so that as P moves, the quantity is always 0.)

Next, draw a point E and construct the line k through E parallel to AD.  Split point P from line AD and merge it to this new line k.  What happens to the quantity ax+by as P moves along the new line?

Three coordinates

Now draw a point F and construct a line m through F parallel to line BC.  Split P from k and merge it with line m.

• Calculate the quantity x + y and check that this is constant on m.

• Calculate z = h[A](P).  Check that x + y = 1 – z and that z = area BCP/area ABC (except possible for sign).
• Calculate x + y + z.  Split P from line m and move P on the plane.  The sum should always be 1.

Examine carefully what happens when P moves outside triangle ABC.  The union of the 3 triangles ABP, BCP, and CAP is larger than triangle ABC, but the regions outside ABC are each covered by two triangles, one of which is positively oriented and one of which is negatively oriented, so that the sum is still one.

Finally, be sure to drag C so that the triangle ABC is oriented in a clockwise direction.  See what happens.  Do the relations still hold?

Optional Tool idea. Make a tool that will take A, B, C and P and will calculate all 3 coordinates x, y, z in one step.

These kinds of questions will appear on assignments and/or tests.

Special Points

• For what points in the triangle is x = 0?
• For what points in the triangle is y = z?
• If x = y = z, where is P?

Centroid Question

In a new sketch draw a triangle ABC and intersect medians to construct the centroid G. What are the coordinates x, y, z for G? How does this follow from a theorem that we know about the centroid?