Polygonal Linkages

Judah L Schwartz, Jan 30, 2016

[i]This is a series of applets drawn from the website [b]mathMINDhabits[/b] [color=#1551b5][sites.google.com/site/mathmindhabits][/color] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/i] All of the applets in this collection approach the exploration of polygons by considering them to be closed chains of hinged links. These applets provide teachers with a myriad of opportunities for posing open-ended problems that can be readily explored. I know that that these applets will keep me [[i]and I hope many teachers and their students[/i]] engaged for a long time to come.

Table of Contents

  1. Point in a polygon
    1. Point in an Equilateral Triangle
    2. Point in a Square
    3. Point in an Equilateral Triangle - revisited
    4. Point in a Regular Pentagon
  2. Exploring polygons
    1. Quadrilateral factory ?
    2. Polygon Explorer
    3. Polygon Synthesizer
  3. Building triangles
    1. Build a triangle from medians?
    2. Build a triangle from Angle Bisectors?
    3. Build a triangle from Altitudes?
  4. Linking Geometry & Algebra
    1. The Kite & The Lemniscate
    2. Parametric polygon explorer
    3. Parametric polygon synthesizer

1.

Point in a polygon

The first of these applets "Point in an Equilateral triangle" deals with a well-known problem - namely the fact that the sum of the distances to the sides of an equilateral triangle from any point in the interior of a triangle is constant. The applet offers a visual proof of this. The second applet "Point in a Square" extends this inquiry to a point in a square. The user may explore under what circumstances the four distances from a point in the interior of the square to its sides or to its vertices can form a closed quadrilateral. The third applet "Point in an Equilateral Triangle - Revisited" and the fourth applet "Point in a Regular Pentagon" are a matched pair. They allow the user to explore the probability that the distances from a point in the interior of the [triangle, pentagon] can be used to form a new [triangle, pentagon].

  • 1. Point in an Equilateral Triangle
  • 2. Point in a Square
  • 3. Point in an Equilateral Triangle - revisited
  • 4. Point in a Regular Pentagon

1.1.

Point in an Equilateral Triangle

The GOLD point is chosen at random inside the equilateral triangle. The red, blue and green segments are lines drawn from the GOLD point and perpendicular to each of the sides. Their lengths vary in size as you move the GOLD point from place to place inside the triangle. However, the sum of their lengths is constant. [br][br]How is the sum of the lengths related to the size of the triangle? Why? Can you prove it?[br][br][br]Would a similar thing be true in a square? Why or why not? [br][br][br]What about other regular polygons with an odd number of sides? with an even number of sides?[br]
– Judah L Schwartz

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2.

Exploring polygons

The applets in this chapter begin with "Quadrilateral Factory?" which extends the previous work on triangles to quadrilaterals. Specifically, it poses the same questions in a setting that is simultaneously less constrained than the triangle but more constrained than the pentagon. The next two applets, "Polygon Explorer" and seventh applet "Polygon Synthesizer", are a matched pair. They introduce a different way of characterizing polygons.

  • 1. Quadrilateral factory ?
  • 2. Polygon Explorer
  • 3. Polygon Synthesizer

2.1.

Quadrilateral factory ?

BREAK A STICK INTO FOUR PIECES.[br][br]Can you [i][b]always[/b][/i] make a closed quadrilateral in the plane using the four pieces?[br]How many quadrilaterals can you make? How do you know?[br][br][br]Can you prove your answer? [br][br][br]Can you make [i][b]any shape[/b][/i] closed quadrilateral in the plane using[br]four sticks the sum of whose lengths is fixed?[br] [br][br]Can you prove your answer?[br][br][br]What closed polygons with more than four sides can you make (and [b][i]not[/i][/b] make) if you break a stick into more than four pieces?
– Judah L Schwartz

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3.

Building triangles

The applets in this chapter, "Build a Triangle out of Medians", "Build a Triangle out of Angle Bisectors" and "Build a Triangle out of Altitudes" invert the usual construction problems - they ask the user to specify three lengths [as medians of a possible triangle, say] and then explore under what circumstances those three chosen lengths can form a triangle.

  • 1. Build a triangle from medians?
  • 2. Build a triangle from Angle Bisectors?
  • 3. Build a triangle from Altitudes?

3.1.

Build a triangle from medians?

Three line segments ([b][i]dotted lines[/i][/b]) intersect at a point that divides the lengths of each of the segments in the ratio of 2 to 1.[br][br]These three line segments could be the medians of a triangle.[br][br]You can use the sliders to set the size of these three line segments. The changed lengths will maintain the 2:1 ratio. Try to set them so that points A and B, B and C, and C and A are joined by straight line segments.[br][br]Alternatively, you can try to drag the points A,B and C to where you think the vertices of the target triangle are. When A, B and C lie on the vertices of the target triangle the angle between two segments of the same color will be 180 degrees.[br][br]Can a triangle always be made this way? Can you prove it?
– Judah L Schwartz

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4.

Linking Geometry & Algebra

The applets in this chapter all present an opportunity to explore the ways in which one can link geometric and algebraic understanding.

  • 1. The Kite & The Lemniscate
  • 2. Parametric polygon explorer
  • 3. Parametric polygon synthesizer

4.1.

The Kite & The Lemniscate

The BLUE kite is made two pairs of two fixed length segments. The BLUE dots are at the midpoints of the kite segments on which they lie.[br][br]One end of the dotted diagonal of the kite is fixed. The GREEN dot at the other end can be dragged along the circle on which it lies.[br][br]What will the paths of the BLUE dots be?
Adapted from an applet by Walerij Koschkin
– Judah L Schwartz

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