Sunday, 28 April 2013

Geometry




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Points, lines, and planes



Points, lines, and planes
This topic introduces the basic conceptual tools that underpin our journey through Euclidean geometry. These include the ideas of points, lines, line segments, rays, and planes.
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Introduction to Euclidean geometry

Roughly 2400 years ago, Euclid of Alexandria wrote Elements which served as the world's geometry textbook until recently. Studied by Abraham Lincoln in order to sharpen his mind and truly appreciate mathematical deduction, it is still the basis of what we consider a first year course in geometry. This tutorial gives a bit of this background and then lays the conceptual foundation of points, lines, circles and planes that we will use as we journey through the world of Euclid.



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Angles and intersecting lines

Angles and intersecting lines
This topic continues our journey through the world of Euclid by helping us understand angles and how they can relate to each other.
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Angle basics and measurement

This tutorial will define what an angle is and help us think about how to measure them. If you're new to angles, this is a great place to start.

Angles between intersecting and parallel lines

Welcome. I'd like to introduce you to Mr. Angle. Nice to meet you. So nice to meet you. This tutorial introduces us to angles. It includes how we measure them, how angles relate to each other and properties of angles created from various types of intersecting lines. Mr. Angle is actually far more fun than you might initially presume him to be.

Angles with triangles and polygons

Do the angles in a triangle always add up to the same thing? Would I ask it if they didn't? What do we know about the angles of a triangle if two of the sides are congruent (an isosceles triangle) or all three are congruent (an equilateral)? This tutorial is the place to find out.

Sal's old angle videos

These are some of the classic, original angle video that Sal had done way back when (like 2007). Other tutorials are more polished than this one, but this one has charm. Also not bad if you're looking for more examples of angles between intersected lines, transversals and parallel lines.



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Congruent triangles


Congruent triangles
If you can take one figure and flip, shift and rotate (not resize) it to be identical to another figure, then the two figures are congruent. This topic explores this foundational idea in geometry.
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Congruence postulates

We begin to seriously channel Euclid in this tutorial to really, really (no, really) prove things--in particular, that triangles are congruents. You'll appreciate (and love) what rigorous proofs are. It will sharpen your mind and make you a better friend, relative and citizen (and make you more popular in general). Don't have too much fun.

Congruence and isosceles and equilateral triangles

This tutorial uses our understanding of congruence postulates to prove some neat properties of isosceles and equilateral triangles.


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Perimeter, area and volume


Perimeter, area and volume
A broad set of tutorials covering perimeter area and volume with and without algebra.
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Perimeter and area of rectangles

How long of a fence do you need? How big is your house? How big is your waistline? What's your hat size? These are fundamentally important questions that need to be answered! This is a tutorial to give you the basics of what perimeter, circumference (really the perimeter of a circle) and area are and then applies the ideas to triangles, rectangles and circles. This is more of review for students who are going through the main geometry narrative and can be skipped if yo u remember it from grade-school.

Perimeter and area of triangles

You first learned about perimeter and area when you were in grade school. In this tutorial, we will revisit those ideas with a more mathy lense. We will use our prior knowledge of congruence to really start to prove some neat (and useful) results (including some that will be useful in our study of similarity).

Triangle inequality theorem

The triangle inequality theorem is, on some level, kind of simple. But, as you'll see as you go into high level mathematics, it is often used in fancy proofs. This tutorial introduces you to what it is and gives you some practice understanding the constraints on the dimensions of a triangle.

Koch snowflake fractal

Named after Helge von Koch, the Koch snowflake is one of the first fractals to be discovered. It is created by adding smaller and smaller equilateral bumps to an existing equilateral triangle. Quite amazingly, it produces a figure of infinite perimeter and finite area!

Heron's formula

Named after Heron of Alexandria, Heron's formula is a power (but often overlooked) method for finding the area of ANY triangle. In this tutorial we will explain how to use it and then prove it!

Circumference and area of circles

Circles are everywhere. How can we measure how big they are? Well, we could think about the distance around the circle (circumference). Another option would be to think about how much space it takes up on our paper (area). Have fun!

Perimeter and area of non-standard shapes

Not everything in the world is a rectangle, circle or triangle. In this tutorial, we give you practice at finding the perimeters and areas of these less-than-standard shapes!

Volume and surface area

Tired of perimeter and area and now want to measure 3-D space-take-upness. Well you've found the right tutorial. Enjoy!



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Similarity



Similarity
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Triangle similarity

This tutorial explains a similar (but not congruent) idea to congruency (if that last sentence made sense, you might not need this tutorial). Seriously, we'll take a rigorous look at similarity and think of some reasonable postulates for it. We'll then use these to prove some results and solve some problems. The fun must not stop!

Old school similarity

These videos may look similar (pun-intended) to videos in another playlist but they are older, rougher and arguably more charming. These are some of the original videos that Sal made on similarity. They are less formal than those in the "other" similarity tutorial, but, who knows, you might like them more.



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Right triangles



Right triangles
Triangles are not always right (although they are never wrong), but when they are it opens up an exciting world of possibilities. Not only are right triangles cool in their own right (pun intended), they are the basis of very important ideas in analytic geometry (the distance between two points in space) and trigonometry.
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Pythagorean theorem

Named after the Greek philosopher who lived nearly 2600 years ago, the Pythagorean theorem is as good as math theorems get (Pythagoras also started a religious movement). It's simple. It's beautiful. It's powerful. In this tutorial, we will cover what it is and how it can be used. We have another tutorial that gives you as many proofs of it as you might need. In thi

Pythagorean theorem proofs

The Pythagorean theorem is one of the most famous ideas in all of mathematics. This tutorial proves it. Then proves it again... and again... and again. More than just satisfying any skepticism of whether the Pythagorean theorem is really true (only one proof would be sufficient for that), it will hopefully open your mind to new and beautiful ways to prove something very powerful.

Special right triangles

We hate to pick favorites, but there really are certain right triangles that are more special than others. In this tutorial, we pick them out, show why they're special, and prove it! These include 30-60-90 and 45-45-90 triangles (the numbers refer to the measure of the angles in the triangle).


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Special properties and parts of triangles

Special properties and parts of triangles
You probably like triangles. You think they are useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined!
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Perpendicular bisectors

In this tutorial, we study lines that are perpendicular to the sides of a triangle and divide them in two (perpendicular bisectors). As we'll prove, they intersect at a unique point called the cicumcenter (which, quite amazingly, is equidistant to the vertices). We can then create a circle (circumcircle) centered at this point that goes through all the vertices. This tutorial is the extension of the core narrative of the Geometry "course". After this, you might want to look at the tutorial on angle bisectors.

Angle bisectors

This tutorial experiments with lines that divide the angles of a triangle in two (angle bisectors). As we'll prove, all three angle bisectors actually intersect at one point called the incenter (amazing!). We'll also prove that this incenter is equidistant from the sides of the triangle (even more amazing!). This allows us to create a circle centered at the incenter that is tangent to the sides of the triangle (not surprisingly called the "incircle").

Medians and centroids

You've explored perpendicular bisectors and angle bisectors, but you're craving to study lines that intersect the vertices of a a triangle AND bisect the opposite sides. Well, you're luck because that (medians) is what we are going to study in this tutorial. We'll prove here that the medians intersect at a unique point (amazing!) called the centroid and divide the triangle into six mini triangles of equal area (even more amazing!). The centroid also always happens to divide all the medians in segments with lengths at a 1:2 ration (stupendous!).

Altitudes

Ok. You knew triangles where cool, but you never imagined they were this cool! Well, this tutorial will take things even further. After perpendicular bisectors, angle bisector and medians, the only other thing (that I can think of) is a line that intersects a vertex and the opposite side (called an altitude). As we'll see, these are just as cool as the rest and, as you may have guessed, intersect at a unique point called the orthocenter (unbelievable!).

Bringing it all together

This tutorial brings together all of the major ideas in this topic. First, it starts off with a light-weight review of the various ideas in the topic. It then goes into a heavy-weight proof of a truly, truly, truly amazing idea. It was amazing enough that orthocenters, circumcenters, and centroids exist , but we'll see in the videos on Euler lines that they sit on the same line themselves (incenters must be feeling lonely)!!!!!!!


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Quadrilaterals



Quadrilaterals
Not all things with four sides have to be squares or rectangles! We will now broaden our understanding of quadrilaterals!
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  1. Quadrilateral Overview
  2. Quadrilateral Properties
  3. Proof - Opposite Sides of Parallelogram Congruent
  4. Proof - Diagonals of a Parallelogram Bisect Each Other
  5. Proof - Opposite Angles of Parallelogram Congruent
  1. Proof - Rhombus Diagonals are Perpendicular Bisectors
  2. Proof - Rhombus Area Half Product of Diagonal Length
  3. Area of a Parallelogram
  4. Whether a Special Quadrilateral Can Exist
  5. Rhombus Diagonals




Circles


Circles
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  1. Language and Notation of the Circle
  2. Circles: Radius, Diameter and Circumference
  3. Parts of a Circle
  4. Area of a Circle
  1. Inscribed and Central Angles
  2. Perpendicular Radius Bisects Chord
  3. Right Triangles Inscribed in Circles (Proof)
  4. Area of Inscribed Equilateral Triangle (some basic trig used)



Angles




Angles
Identifying, measuring, and calculating angles.
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  1. Angle basics
  2. Measuring angles in degrees
  3. Using a protractor
  4. Measuring angles
  5. Acute right and obtuse angles
  6. Complementary and supplementary angles
  7. Identifying Complementary and Supplementary Angles
  8. Acute Obtuse and Right Angles
  9. Angle-a-trons
  10. Angles (part 2)
  11. Proof - Sum of Measures of Angles in a Triangle are 180
  12. Triangle Angle Example 1
  13. Angles at the intersection of two lines
  14. Proof-Vertical Angles are Equal
  15. Angles (part 3)
  16. Angles formed between transversals and parallel lines
  17. Proof - Corresponding Angle Equivalence Implies Parallel Lines
  18. Angles of parallel lines 2
  1. Triangle Angle Example 2
  2. Triangle Angle Example 3
  3. Finding Missing Angles
  4. Finding more angles
  5. The Angle Game
  6. Angle Game (part 2)
  7. More on why SSA is not a postulate
  8. Angles Formed by Parallel Lines and Transversals
  9. Point Line Distance and Angle Bisectors
  10. Two column proof showing segments are perpendicular
  11. Angle Bisector Theorem Examples
  12. Angle Bisector Theorem Proof
  13. Inscribed and Central Angles
  14. Perpendicular Radius Bisects Chord
  15. Inradius Perimeter and Area
  16. 2003 AIME II Problem 7
  17. Introduction to angles (old)



Triangles



Triangles
Identifying types of triangles, using principles of similarity and congruence, and applying and proving triangle postulates.
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  1. Proof - Sum of Measures of Angles in a Triangle are 180
  2. Triangle Angle Example 1
  3. Triangle Angle Example 2
  4. Triangle Angle Example 3
  5. Challenging Triangle Angle Problem
  6. Congruent and Similar Triangles
  7. Congruent Triangles and SSS
  8. SSS to Show a Radius is Perpendicular to a Chord that it Bisects
  9. Other Triangle Congruence Postulates
  10. Finding Congruent Triangles
  11. More on why SSA is not a postulate
  12. Congruent Triangle Proof Example
  13. Congruent Triangle Example 2
  14. Congruent legs and base angles of Isosceles Triangles
  15. Equilateral Triangle Sides and Angles Congruent
  16. Equilateral and Isosceles Example Problems
  17. Another Isosceles Example Problem
  18. Example involving an isosceles triangle and parallel lines
  19. Figuring out all the angles for congruent triangles example
  20. Triangle Area Proofs
  21. Area of an Equilateral Triangle
  22. Koch Snowflake Fractal
  23. Area of Koch Snowflake (part 1) - Advanced
  24. Area of Koch Snowflake (part 2) - Advanced
  25. Similar Triangle Basics
  26. Similarity Postulates
  27. Similar triangles
  28. Similar triangles (part 2)
  29. Similar Triangle Example Problems
  30. Similarity Example Problems
  31. Application of Similar Triangles
  32. Challenging Similarity Problem
  33. Similarity example where same side plays different roles
  34. Finding Area Using Similarity and Congruence
  35. Introduction to the Pythagorean Theorem
  36. The Pythagorean Theorem
  37. Pythagorean Theorem II
  1. Pythagorean Theorem 1
  2. Pythagorean Theorem 2
  3. Pythagorean Theorem 3
  4. Garfield's proof of the Pythagorean Theorem
  5. Bhaskara's proof of Pythagorean Theorem
  6. Pythagorean Theorem Proof Using Similarity
  7. Another Pythagorean Theorem Proof
  8. 45-45-90 Triangles
  9. 45-45-90 Triangle Side Ratios
  10. Intro to 30-60-90 Triangles
  11. 30-60-90 Triangles II
  12. 30-60-90 Triangle Side Ratios Proof
  13. 30-60-90 Triangle Example Problem
  14. Heron's Formula
  15. Part 1 of Proof of Heron's Formula
  16. Part 2 of the Proof of Heron's Formula
  17. Area of Inscribed Equilateral Triangle (some basic trig used)
  18. Right Triangles Inscribed in Circles (Proof)
  19. Area of Diagonal Generated Triangles of Rectangle are Equal
  20. Circumcenter of a Triangle
  21. Circumcenter of a Right Triangle
  22. Incenter and incircles of a triangle
  23. Medians divide into smaller triangles of equal area
  24. Exploring Medial Triangles
  25. Proving that the Centroid is 2-3rds along the Median
  26. Median Centroid Right Triangle Example
  27. Proof - Triangle Altitudes are Concurrent (Orthocenter)
  28. Review of Triangle Properties
  29. Euler Line
  30. Euler's Line Proof
  31. Common Orthocenter and Centroid
  32. Basic Triangle Proofs Module Example
  33. Basic Triangle Proofs Module Example 2
  34. Fill-in-the-blank triangle proofs example 1
  35. Wrong statements in proofs example 1
  36. Fill-in-the-blank triangle proofs example 2



Worked Examples



Worked Examples
Sal does the 80 problems from the released questions from the California Standards Test for Geometry. Basic understanding of Algebra I necessary.
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  1. Interesting Perimeter and Area Problems
  2. Challenging Perimeter Problem
  3. CA Geometry: deductive reasoning
  4. CA Geometry: Proof by Contradiction
  5. CA Geometry: More Proofs
  6. CA Geometry: Similar Triangles 1
  7. CA Geometry: Similar Triangles 2
  8. CA Geometry: More on congruent and similar triangles
  9. CA Geometry: Triangles and Parallelograms
  10. CA Geometry: Area, Pythagorean Theorem
  1. CA Geometry: Area, Circumference, Volume
  2. CA Geometry: Pythagorean Theorem, Area
  3. CA Geometry: Exterior Angles
  4. CA Geometry: Deducing Angle Measures
  5. CA Geometry: Pythagorean Theorem, Compass Constructions
  6. CA Geometry: Compass Construction
  7. CA Geometry: Basic Trigonometry
  8. CA Geometry: More Trig
  9. CA Geometry: Circle Area Chords Tangent
  10. CA Geometry: Secants and Translations














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