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Brilliant

Geometry I

via Brilliant

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Overview

In this course, you'll solve delightful geometry puzzles and build a solid foundation of skills for problem-solving with angles, triangles, polygons, and circles. You'll learn how to come up with clever, creative solutions to tough challenges and explore a wide range of theorems.

This course is the perfect place to start (or continue) your exploration of geometry if you know how to measure angles and calculate the areas of rectangles, circles, and triangles; and want to learn the next level of geometric problem-solving techniques.

Additionally, this is a great course to take if you want to strengthen your geometric intuition in preparation for taking a geometry or design course in school. You'll also need to use a little bit of fundamentals-level algebra in this course, but nothing more advanced than two-variable equations, squares, and square roots.

Syllabus

  • Mathematical Thinking
    • Pythagoras' Theorem: A visual proof of the most famous theorem in all of mathematics.
    • The Distributive Property: If you replace algebra with geometry, you'll never need to factor again.
    • Difference of Squares: This isn't how you learned this identity at school.
    • Difference of Squares Challenges: Put all your learnings to the test.
    • The Quadratic Formula: Use visual intuition and algebra to solve every quadratic ever.
  • Introduction
    • Triangles and Hexagons: Get started by thinking about patterns that combine triangles and hexagons.
    • Strategic Geometry: Work out strategies to solve these area challenges.
    • Driving on a Polygon: Derive a fundamental theorem of geometry!
  • Angles
    • Angle Hunting Axioms: Practice finding missing angles.
    • Advanced Mental Shortcuts: Learn sneaky strategies for solving tough angle problems.
    • Internal Angles in a Polygon: Solve angle problems involving polygons that have many sides.
    • Invariant Angle Sets: Investigate scenarios where the sum of several angles stays constant.
    • Advanced Angle Hunts: Tackle this last set of angle challenges by combining all of the techniques you've learned so far.
  • Triangles
    • The Triangle Inequality: In what circumstances can you make a triangle?
    • Congruent and Similar Triangles: When are triangles the same?
    • Bass Fishing: Investigate a special case where a triangle can be drawn two ways.
    • Curry's Paradox: Uh... Where did the missing square go?
  • Composite Polygons
    • Composite Figure Warm-ups: Warm up your problem-solving muscles by mixing and matching bits of different shapes.
    • Adding Lines and Grids: Add lines to help clarify your thinking.
    • Complementary Areas: Sometimes it's easier to find the area of what's left after your shape is removed...
    • Inclusion and Exclusion: What happens when shapes overlap?
    • Invariant Areas: When bend, stretch, move and spin parts of these figures, the areas change shape but don't change size.
  • Regular Polygons
    • Angles of Regular Polygons: When all of the sides are the same length and all of the angles are the same measure, what else must be true?
    • Is It Regular?: Can you be absolutely certain that each of these polygons is regular?
    • Polygon Areas and Lengths: Apply the Pythagorean Theorem to find length and area measures of regular polygons.
    • Matchstick Polygons: Play around with these polygons made out of matchsticks, q-tips, and toothpicks.
    • Stellations: By extending all of the edges of a polygon you can make beautiful stars.
    • Dissections: Practice your skills polygon problem-solving with these dissected polygon puzzles.
  • In and Out of Circles
    • Central Angles and Arcs: Investigate geometric patterns using central angles.
    • Thales's Theorem: What happens when an angle is inscribed in a semicircle?
    • Inscribed Angles: Extend Thales' observation into another beautiful and more general theorem.
    • Puzzles with Inscribed Angles: Warm up with this round of practice problems that explore the Inscribed Angle Theorem.
    • Cyclic Quadrilaterals: Study the properties of quadrilaterals inscribed inside of circles.
  • Secants and Tangents
    • Power of a Point I: Intersecting lines inside a circle are a special circumstance worth investigating in detail!
    • Intersecting Secants: What happens when lines intersect inside a circle?
    • Power of a Point II: Solve a problem assortment that puts all of your new theorems to use!
    • Tangents: What happens when lines just barely touch the outside of a circle?
    • Problem-Solving Challenges: Practice and strengthen the entire set of tools you've learned so far with this final round of challenges.
  • Mastering Triangles
    • Right Triangles: Start your journey into advanced triangles on the right (aka 90-degree) foot.
    • Thales + Pythagoras: Combine what you know about Thales and Pythagoras to approach some fascinating problems.
    • Cevians: Explore what happens when you connect up one point and one side.
    • Pegboard Triangles: What happens when the triangles are drawn on a regular grid?
  • Triangle Centers
    • Three Different Centers: Learn about the three most commonly used triangle centers and explore how they relate to each other.
    • The Circumcenter: Use perpendicular bisectors to experiment with a fourth type of "center."
    • Euler's Line: Prove a profound result that relates the orthocenter, centroid, and circumcenter.
    • Morley's Triangle: Use trisectors rather than bisectors to get an astonishing result.
  • When Geometry Gets Tough
    • Geometric Stumpers: Challenge yourself and use any strategies you want to solve these problems!
    • Challenging Composites: Extend your best strategies even further and look for shortcuts.
    • Coordinate Geometry: Return to the coordinate plane for some especially challenging coordinate puzzles.
    • Advanced Angle Hunting: Go far beyond the first axioms and explore a realm where wild geometry flourishes.
    • Applying the Pythagorean Theorem: Apply your mastery of triangles to this final exploration!
  • Beautiful Geometry
    • Infinite Areas: How can you fit infinitely many shapes in a finite space?
    • Polyomino Tiling: Get your first taste of beautiful geometry by exploring these tiling puzzles.
    • Guards in the Gallery: Place guards so that they can see into every corner of these irregular polygons.
  • Tessellations and Reptiles
    • Regular Tessellations: Explore tessellation patterns that use only one type of regular polygon.
    • Semiregular Tessellations: What tessellations can you make when you use multiple types of shapes?
    • Transforming Tiles Part 1: Morph regular shapes into bizarre shapes to create new kinds of tilings.
    • Transforming Tiles Part 2: Extend your knowledge and skills with a second round of transformation puzzles.
    • Irregular Tiles: Some very strange shapes can tile the plane, but the tessellations they make can be truly bizarre.
    • Reptiles: Make larger copies from smaller copies.
    • Infinite Arithmetic: Learn how geometric series can be summed up geometrically!
  • Polyominos
    • Tiling a Chessboard: In which of these cases can you entirely cover the chessboard with dominos?
    • Counting All Possible Solutions: How many different ways do they fit?
    • Bigger Polyomino Blocks: Take a step beyond dominoes and tackle these tetrominoes and pentominoes challenges!
    • Challenging Packing Puzzles: Apply insight and creativity to pack these polyominos as tightly as possible.
    • X-Only: What if you can only use the X-shaped pentomino?
    • Tiling and Cutting: Investigate methods for finding out if a tiling is possible.
    • Congruent Cutting: Cut shapes into several pieces that are identical.
  • Folding Puzzles
    • Mathematical Origami: Unfold a paper crane and study the mountains and valleys that the folds reveal.
    • Dragon Folding: To make a dragon fractal, you just have keep on folding, and folding, and folding...
    • 1D Flat Folding: Explore the rules that govern how a single piece of paper can be folded flat.
    • 2D Holes and Cuts: Fold, then cut, and then unfold again to make these designs.
    • 2D Single-Vertex Flat Folding (I): Mathematically, how can you tell if something is flat foldable?
    • 2D Single-Vertex Flat Folding (II): Extend your exploration of flat folding one final step further.
  • Guarding Galleries
    • Strange Polygons: Get acquainted with the unusual polygons found in art gallery puzzles.
    • Convex vs. Concave: Study the difference between convex and concave shapes and how they affect guard placement.
    • Quadrilateral and Pentagonal Galleries: Look specifically at cases in which the galleries are quadrilaterals and pentagons.
    • Efficient Guard Placement: Is there a systematic strategy for finding an ideal guard placement?
    • Worst-Case Designs: Practice making galleries that are tough to guard.
    • Fisk's Coloring Proof: Fisk's proof is puts an upper bound how many guards you might need for an n-sided gallery.
    • Further Art Gallery Research: Investigate internal walls and other twists that you can add to the art gallery puzzle.
  • Pick's Theorem
    • Pegboard Rectangles: Begin studying Pick's Theorem with an intuitive case.
    • Pegboard Triangles: What happens when you cut pegboard rectangles in half?
    • Pick's Theorem Generalized: Prove Pick's Theorem for any lattice polygon.
    • Pick's Theorem with One Hole: Poke a hole in your polygons and see what formula comes out.
    • Pick's Theorem with Multiple Holes: Extend Pick's Theorem one more time to address this multi-holed variation.

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