# Construction Pod Game: Part D

Topic:
Geometry

## Welcome to the Construction Pod Game

The Construction Pod Game is a series of challenges for your pod to construct interesting and fun geometric figures. The Construction Pod Game is divided into five Parts. This is Part D. If your pod has not yet completed Part C, please go to Construction Pod Game: Part C. Put your Construction Crew Pod together again with three, four, five or six people from anywhere in the world who want to play the game together online. Collaborate, share ideas, ask questions and enjoy.

## LEVEL 9: TRANFORMATION LEVEL

This level will explore the idea of deductive proof in geometry. This was the great discovery in mathematics, that you could show by careful argument why something had to be true. In particular, a set of theorems about congruent triangles are very handy for proving many things in geometry. Understanding them will let you tackle some difficult challenges about inscribed polygons.

## Challenge 30: Translate by a Vector

See the menu item for reflection about a line (the diagonal line with a blue point on one side reflected by a red point on the other). There are several GeoGebra tools for geometric "transformations". Try these tools out in this set of five Challenges.

## Challenge 31: Reflect About a Line

1. Could you reverse the two reflections to get back to the original position? 2. Could you find another path of reflections to get back to the original position? 3. Can you translate either of the reflections back to the original? 4. Are the three triangles congruent to each other? Could you lay them on top of each other by translating them around? 5. If you reflect ABC about line DE and then about line EF does that have the same result as reflecting ABC about line EF and then about line DE?

## Challenge 32: Rotate Around a Point

How are rotations different from translations? Drag point D and then describe how ABC is rotated. Does the order of the two rotations matter? Would the final triangle be the same if ABC was first rotated about point E and then about point D? Give an example of a reflection of ABC followed by a translation that would end up the same as A'B'C'.

## Challenge 33: Combine Transformations

Describe the transformations you did. Did you have any trouble doing the different tasks? Can you replace every translation with a series of reflections and rotations? Can you replace every reflection with a series of translations and rotations? Can you replace every rotation with a series of reflections and translations?

## Challenge 34: Create Dynamic Patterns

Can you make dynamic patterns of triangles using a repeated rotation or a repeated reflection? Then drag points to move the pattern in interesting ways.

In this level, you will explore four-sided figures. There are many more possibilities with four sides than with just three.

## Challenge 35: Construct Quadrilaterals with Constraints

What constraints do you think were constructed into poly1? What constraints do you think were constructed into poly2? What constraints do you think were constructed into poly3? What constraints do you think were constructed into poly4? Were you able to construct your own quadrilateral with the same constraints as one of the original ones? Did you drag it to make sure it had the same behavior?

## Challenge 36: Construct a Rhombus

Describe the steps you used to construct a rhombus using circles. Describe the steps you used to construct a rhombus using reflections. Describe another way to construct a four-sided figure with equal side lengths (a regular quadrilateral or a "rhombus").

Were you surprised about the relation of the areas of the inscribed quadrilateral to the inscribing (exterior) quadrilateral? The areas are displayed in the figure and change as you drag the vertices.) Were you surprised about the constraints on the inscribed quadrilateral being different from those on the inscribing (exterior) quadrilateral? Did you notice the relationship of opposite sides and of opposite angles? The proof of these features of the inscribed quadrilateral is complicated. You probably do not know enough theorems to prove it yourself. Are you able to follow the argument in the proof outlined in the hint?

## Challenge 38: Build a Hierarchy of Quadrilaterals

Do you understand this diagram of constraints or dependencies? For instance, a square is a quadrilateral with all of the constraints: each of its angles is a right angle and each of its side lengths is dependent on the first side length. A rectangle is not constrained to have all its side lengths equal, but it must have two pairs of equal length sides (opposite each other) and four right angles. Can you make a diagram of this same hierarchy with the names of figures (like square, rhombus, kite, parallelogram, etc.) instead of the descriptions of constraints? ("Quadrilateral", "rectangle" and "square" are already shown.) Are there some possible figures that do not have names? Are there some more possible combinations of constraints that could be added to the diagram? In Challenge 15, you constructed an isosceles-right triangle. Can you construct an isosceles-right quadrilateral now (with two equal sides and one right angles)? Where would it go in the diagram? Do you see how the diagram shows that all squares are rectangles? Do you see how the diagram shows that a rectangle can be a square, but it does not have to be?

## Continue to "Construction Pod Game: Part E"

Part E starts on Level 11: Advanced Geometer Level. Congratulations on mastering Part D. You now understand some of the most important methods of transforming theorems about geometry figures and working with quadrilaterals. Part E, presents challenges for advanced students, who have completed all the previous Parts.