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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, and polygons. You'll also improve your visual intuition and learn how to come up with clever, creative solutions to tough challenges.

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

  • Introduction: Get started with 2D on a journey with polygons and clever geometry puzzles.
    • 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: Beware: even a cute little angle can bite!
    • 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: Endless complexity can come from just three sides.
    • 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: Celebrate the surprising truths inherent to the mathematics of space.
    • 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: From perimeters and pi to stellations and tessellations.
    • 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.
  • When Geometry Gets Tough: Apply what you’ve learned to these challenging problems.
    • 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!

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