("the complete quadrangle experiment") draw a line ` and label three points a, c, b on this line. (it works better if c lies somewhere between a and b but this is not necessary). draw two new lines passing through a and call them m1 and m2. draw a third new line through b which intersects both m1 and m2 and call it n. set m1 ∩ n = p and m2 ∩ n = q. draw ←→ cp and label ←→ cp ∩m2 = r. draw ←→ cq and label ←→ cq ∩m1 = s. (p, q, r, s are called a "complete quadrangle".) draw ←→rs and label ←→ rs ∩` = d. repeat this process several times, each time starting with the same positions for a, b and c on `, but using different lines m1, m2 and n. what do you observe? can you think of any reasons that justify your observations? we will study why what you found occurs later in the course.

  Point D is invariant

Step-by-step explanation:

A geometry tool such as GeoGebra can help with this effort.

Using the geometry tool, I found it easier to move points P and Q, rather than trying to move lines m1, m2, and n. That has the same effect.

The problem setup has points A, B, and C remaining where they started. I was a little surprised to see that point D also remains in the same location. (I could not see any obvious reason why. Perhaps proportions are involved.)

Attached are two different sets of m1, m2, and n for the same A, B, C. You can see that point D remained in the same place. "Quadrangle" PQRS is colored in both figures.

5x+40

5(x + 8)

7x - 49

7(x - 7)

18x + 9

9(2x + 1)

15x - 10

5(3x - 2)

step-by-step explanation:

for number one all you have to do is factor 5 out of 5x + 40

for number 2 factor 7 out of 7x - 49

for number three factor 9 out of 18x + 9

for number four factor 5 out of 15x - 10