Construction Pod Game: Part B

What is constrained for each of these triangles: poly1, poly2, poly3, poly4 and poly5? Drag each vertex point to see if you can change the type of angle or the relationships of the sides. Can you drag poly1 and each of its points so that it exactly covers any of the other triangles? Can you drag any other triangle and each of its points so that it exactly covers any of the other triangles? Can you name the type of each triangle?

Did you figure out how to do this challenge without looking at the hint? Did you think about the definition of a circle, where all radii are equal length? Can you drag your isosceles triangle to look like a right triangle or an equilateral triangle? How do you think about the fact that it is always isosceles, but can sometimes look (or even measure) right or equilateral?

Did you use the perpendicular tool or did you construct the perpendicular to your base segment going through one of its endpoints (like in Challenge 10)? Remember that a right angle measures 90 degrees. Can you construct a figure that combines two right triangles and shows that a straight line is an angle of 180 degrees? Can you construct a figure that combines four right triangles and shows that a circle has 360 degrees?

Did you need the hints to do this? It is interesting to you that one figure can have more than one dependency built into it? Why would this be a powerful idea? Now you can combine multiple dependencies in one figure or multiple figures (like four right-isosceles triangles) in one larger figure (like a square) with many dependencies.

Can you do this whole construction? Can you even follow it step-by-step? Imagine the ancient Greeks who invented geometry thinking up this complicated procedure. This method of copying a length is presented in the beginning of Euclid's book, because it is needed for many other constructions and proofs. It is preceded by the method for constructing an equilateral triangle (which you did in Challenge 7), because that is used in this method. Did you ever hear that "equality is transitive." That means that if A=B and B=C then A=C. Euclid use this to construct a long series of equal length segments to prove that the length of the final segment CH is equal to the length of the original segment AB. The equalities are based on the fact (or definition or axiom) that all radii of the same circle are equal length segments. Drag points A, B or C to see how the length of AB is copied no matter where these points are.

Using GeoGebra's compass tool is like using a physical compass (or caliper). You put one end at point A and one end at point B to set a span of length AB. Then move the compass to a desired point C. The other end of the compass can then be put anywhere on a circle around point C of radius AB. What happens to segment CE when you drag segment AB or one of its points? Next time you want to transfer a segment length, will you use the compass tool or do the construction from Challenge 16?

In this challenge, you can see the difference between copying a length to a new segment (so that the new version is still dependent on the original segment) and using copy-and-paste to make a static copy of a length, which is not dependent on changes of the original segment. Which points, segments or circles are free to be dragged without constraint? Which are completely dependent and can only be moved indirectly be dragging another point that it is dependent upon? Are there any that can be moved somewhat, but only in a constrained way?

For this challenge, the lengths of some segments are shown. You can show the length of a segment by selecting the segment with the arrow tool and then going to the menu item "Object Properties." Check the box for "Show Label" and select "Value" for the label. In geometry, you never really have to measure lengths or angles -- you just construct them to have the values you want. But it is sometimes reassuring to show their measures when you are learning with GeoGebra. What segment has a length equal to the lengths of what other segments in the example? Were you able to construct a segment whose length is equal to the lengths of two other segments? Can you construct a triangle and then construct a line segment whose length is equal to the sum of the lengths of the three sides of the triangle? Does is still work when you drag the vertices of the triangle?

Were you able to make both kinds of copies of your triangle? Did you have any problems or discover any tricks? Describe in your own words the difference between copying with copy-and-paste versus copying with the compass tool.

Did you understand how to copy the angles? To copy an angle like BAC to a new angle like HDI requires two copies of lengths using the compass tool. First, use the compass to measure out from vertex A to some distance (like AF) out one of the sides (it does not matter what distance out). Then copy the distance to vertex D, creating DH, which equals AF. Also mark points G and I, where the compass crosses the other sides of the angles at A and D. Now use the compass tool to copy the distance FG to H and mark point I where the two circles for the compass lengths cross and construct a ray from point D going through point I. Now lengths AF = AG = DH = DI. The new angle HDI is the same size as angle FAG because the distance between the two sides of each angle is an equal length at the same distance out the sides. GeoGebra does not have a tool for copying angles. You have to construct the equal angle using the compass tool. Do you understand how to construct a triangle "similar" to triangle ABC? Summarize in your own words how to construct a similar triangle by copying the three angles. Work with your team mates in your pod to write a brief proof of how you know the new triangle is similar to the original one.