Basic Constructions
Introduces basic constructions and takes students through several different investigations in which they can apply their newly learned skills!
Table of Contents
- Definitions
- What's Your Point?!
- I Walk The Line
- Leaving on A Jet Plane
- Angles and Demons
- How Do You Figure?
- Circling The Drain
- Is Round "In Shape?"
- Is Bermuda One?
- Square Jordans, Shoes for Nerds
- I'm Beside Myself
- Postulates
- What is A Postulate?
- Postulate 1
- Postulate 2
- Postulate 3
- Postulate 4
- Postulate 5
- Common Notions
- What are Common Notions?
- Common Notion 1
- Common Notion 2
- Common Notion 3
- Common Notion 4
- Common Notion 5
- Basic Constructions
- Rules of Construction
- Copy a Line Segment
- Perpendicular Bisector of a Line
- Perpendicular Line Through a Point
- Bisecting an Angle
- Copying an Angle
- Parallel Line through a point
- Equilateral Triangle
- Intermediate Constructions
- Inscribe a Circle in a Square
- Altitudes of Triangles
- Medians of a Triangle
- Perpendicular Bisectors of a Triangle
- Angle Bisectors of a Triangle
- Challenge Constructions
- Construct a Rhombus
- Finding the Center of a Circle
- Construct a 45 Degree Angle
- Construct a 30 Degree Angle
- Expert Constructions
- Inscribe a Circle in a Rhombus
- Finish the Triangle 1
- Finish the Triangle 2
- Three Equal Lines
- Tangent Circle Through Two Points
- Master Construction
- The Nine Point Circle
What's Your Point?!
Definition 1
[b]A [i]point[/i] is that which has no part.[/b][br][br]In other words, [u]a point is a location without dimension[/u]. How wide is a point? Seems like a weird question, but it is important. A point has no width, no depth, no height; it is just marking a place on the paper. If I asked you to draw a point on a paper with your pencil you would probably make a small dot which obviously has a width that you can measure but the idea of a point is that it is a "dot" that you could not measure. You could zoom in or out of the paper and the dot would not get any bigger or smaller.[br][br]Use the [icon]/images/ggb/toolbar/mode_point.png[/icon] below to make points below (very exciting, it's all we have right now) and zoom in or out, the point doesn't change size because we view it as not having a physical size, it is just saying "this place right here."[br][br]The [icon]/images/ggb/toolbar/mode_move.png[/icon] can be used to drag the screen around, [icon]/images/ggb/toolbar/mode_zoomin.png[/icon][icon]/images/ggb/toolbar/mode_zoomout.png[/icon] let you zoom in and out by clicking on the screen, and the trash can lets you erase. Always click on [icon]/images/ggb/toolbar/mode_move.png[/icon]when you aren't drawing anything, otherwise it will keep drawing when you don't mean to.
What is A Postulate?
Euclid's meaning of a postulate is a little different than what we tend to mean in modern times. Today, postulates and axioms mean the same thing now, a statement that does not require proof; Euclid's means a true statement that may not be initially obvious.[br][br]We will go over true axioms in the next chapter and focus on Euclid's style of postulate now. Think of this section as the slightly more advanced tutorial while the definitions were the basics. Think about the tutorial in a video game, after they show you what each button does you learn combinations and mechanics that are less obvious , that's what this section is about![br][br]There are five basic postulates that we will cover individually, read through them and see if you can explain them in your own words what each means:[br][br][b]1. To draw a straight line from any point to any point.[br][br]2. To produce a finite straight line continuously in a straight line.[br][br]3. To describe a circle with any center and radius.[br][br]4. That all right angles equal one another.[br][br]5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.[/b][br]
What are Common Notions?
In other words "axioms"
In this section we actually get to see true axioms (postulates) in the way we define the word today. An axiom is a statement that is true without proof; the truth in an axiom can come in different ways. [br][br] For example, you can have logical axioms like "A statement and it's opposite cannot both be true simultaneously." A door is either open or closed (we will consider "closed" to be completely latched and anything other than that will be "open" even if you can't actually fit through the door), so we can logically say that a door cannot be open and closed at the same time. We don't have to prove this, we know it and understand it from how we defined our terms and by using basic logic.[br][br] The second type of axiom is non-logical and this is where things get interesting for math. For this section I will keep it simple but if you like to read more of my ramblings, just see the next section below. For non-logical axioms these are statements that are true because we say that they are, they don't follow from any source of logic or common sense but they must be true. In fact, axiom comes from a Greek word that means "to deem worthy." These are the foundation stones from which we build mathematics. For example, lets look at some of the Peano Axioms for natural numbers:[br][br]1. 0 is a natural number[br]2. For every natural number x, x=x. Equality is reflexive[br]3. For all natural numbers x and y, if x=y, then y=x. Equality is symmetric[br]4. For all natural numbers x, y, and z, if x=y and y=z, then x=z. Equality is transitive[br][br] Those words are probably familiar from middle school pre-algebra and you probably though this stuff was common sense, of course a number must equal itself, right? But why? Why is 0 a natural number? It doesn't have to be, in fact when these were written it didn't say 0, it said 1 instead and it worked just fine, we changed it later just for consistency. These statements create a basic framework for math in the natural numbers, they aren't true because the universe says they are, they are true because we want them to be and they make this thing we created called arithmetic systematic and we have found that arithmetic is very useful for our lives. Math that is useful to solve problems we tend to keep around, while not so useful math tends to fade away (look up Lunar Math). We could imagine a system were the above statements are completely different, no problem, the question would then be "Are these new statements useful?" Sometimes it can take centuries for that question to be answered and one new development makes what seemed like an insane idea become the cornerstone for a new type of mathematics.[br][br] For our purposes, we will be using non-logical axioms in this section.
Ramblings...
If ever a mantra described math in my mind, it would be "Reality can be whatever I want." [br][br] Most people are familiar with the scientific method: Identify a problem, form a hypothesis, experiment, analyze data, adjust hypothesis, repeat. science is our way of explaining our universe and science that does not explain what we see and experience is not useful and we adjust our understanding or completely abandon the idea. For centuries people believed that odors caused disease (if you ever wondered why plague doctors are depicted with those long bird masks, it was a breathing tube filled with sweet smelling herbs to "kill" the odor and prevent disease) but when we studied the problem and put those ideas to the test, it fell apart and we realized that microorganisms not only existed but had the power to kill. Science must come from reality and we write laws to say what things happen and theorems to explain why things happen. [br][br] Math... Math is a different story. Science is discovered and math is created, like art. There is nothing about the universe that we studied to find math, it doesn't exist, even numbers are made up. Think of five, what do you see? The symbol 5? Maybe the word "five"? What if your language doesn't use the Latin alphabet? What if I asked you to show me five, how would you do it, with an open hand? But that's not five... yes it's five [u]fingers[/u] but I asked for the thing that is [i]five. [/i]I always enjoy teaching complex (nonreal) numbers to students and listening to the "why do we have to learn about something that's not real!?" and then asking the questions I just asked you about 5. Why do we accept 5 as "real" when it doesn't seem to exist while 4+2i is banned to the realm of "nonreal." Now I do want to clarify, math is real, it is just not a physical thing, it's an idea. It can be whatever we need it to be in this moment (as long as it is logical) but the beautiful thing is (and yes, math is incredibly beautiful) that when we develop math on our own, completely in our minds we can typically find a use for it in the real world but sometimes we can't and that's fine, it's still real.[br][br] Science is like a tool maker looking for problems then designs a tool to solve them. If the tool doesn't work then it either throws it away or modifies it until it does work. Math is like a tool maker who sits in his workshop all day and makes random tools, knowing one day, someone will be smart enough to figure out what problems they solve.
Rules of Construction
What is a Construction?
A geometric construction is a precise drawing that historically only uses a compass and an unmarked straightedge (not a ruler) to solve geometric problems. Ideally, our paper is considered infinite, our lines are also infinitely long and we are not allowed to measure distances. You would be shocked the kinds of math you can do with just lines and circles.[br][br]We describe figures in our drawings in two different ways, given or constructed. A [b]given[/b] figure is either provided at the start of the problem or arbitrarily picked while a [b]constructed[/b] figure is drawn using the rules of construction and is considered precise.
Rules of Construction
[b]Tool Rule:[/b] The only allowed tools are a compass and an unmarked straightedge[br][br][b]Point Rule:[/b] A point must either be given or constructed in the following ways:[br]1. The intersection of two lines[br]2. The intersection of a line and a circle[br]3. The intersection of two circles[br][br][b]Straightedge Rule:[/b] A straightedge can connect two points to construct a line, lines can be infinite[br][br][b]Compass Rule:[/b] A compass can construct a circle with a center and a point on its circumference. A compass can also copy a length
Tool Practice
Watch the animation below of me constructing a square with just these rules, feel free to use the applet below that to try and copy my steps. Don't worry if it seems overwhelming right now, we will work up to this.
Inscribe a Circle in a Square
In the given Square, construct a circle which is [b]inscribed[/b]. Inscribed means that the circle touches the edges of the square at exactly one place on each side; see the image below. Your construction should pass the "drag test" which means if I move your blue points around, your picture should stay correct![br][br]If you get stuck, use the video to help you along but try to do these on your own! There are multiple ways to construct the circle!
Construct a Rhombus
Challenge Construction 1
In the given rectangle, construct a rhombus that uses A and D as vertices. Use the measurement tool when you are done to show you have constructed a rhombus. Return to the definition section if you need a reminder of what a rhombus is.[br][br]Your figure must pass the "drag test."[br]
Inscribe a Circle in a Rhombus
Expert Construction 1
Inscribe a circle in the rhombus provided, must past the "drag test!"
The Nine Point Circle
Construct the Nine Point Circle, as shown and described below; must pass the "drag test."[br][br]1. In triangle ABC construct the mid points of the three sides[br]2. In triangle ABC construct the altitudes of the three sides; mark the base of the altitudes where it meets the sides of the triangle and the orthocenter[br]3. In triangle ABC construct the midpoints of the distance between the orthocenter and each of the vertices[br]4. In triangle ABC construct the circumcenter[br]5. In triangle ABC construct the midpoint between the circumcenter and the orthocenter; call this point N[br]6. Construct a circle with its center at N and a point on the circumference at the midpoint of one of the sides of triangle ABC; this is the Nine Point Circle