Shifty Shapes

Your study of geometry should not rely on tedious, boring memorization of facts and formulas. Mathematics is far more than that. Especially in geometry, your intuition and abilities to make logical deductions should be your chief guides. You may be surprised by how much you can figure out about a shape or diagram, starting from the facts presented in the problem and then expanding what you know by logical deduction. Draw your own diagrams and play around with the shapes, looking for symmetries and patterns. As a bonus, geometric diagrams can be quite beautiful!

Here are some tips to get you started:

1. Read through the entirety of the problem carefully before you start working, and take careful note of any special conditions imposed on the shapes or transformations that the problem describes.
2. Draw out diagrams for yourself, labeling your sketches with all of the information from the problem.
3. Add information to your sketches that you can deduce using what you know of the properties of lines, angles, and specific shapes. Don't be afraid to extend lines and add more shapes and measurements to your diagrams.
4. Symmetry is at the heart of many geometry puzzles. There are three basic kinds of symmetry: reflection (flipping), rotation (spinning), and translation (sliding) that play key roles in many geometry problems.
5. Be careful to not confuse assumptions that rely on features in your sketches with deductions that can be made for certain from the facts that guided making those images. A rough sketch (or even a carefully computer-drawn diagram) can make two lines or angles look approximately equal, when in fact they're not equal at all.

Can we use each of the shapes on the left exactly once to make the shape on the right without any cutting?

Note that the problem specifies that the side lengths of the hexagon are the same as the side lengths of the small triangle. So if we slide a small triangle to fit against the hexagon what happens?

Yep! An equilateral triangle extends the sides of the hexagon. And because of the symmetry of the hexagon, we can repeat this for alternating sides around the hexagon to obtain

Back to Quiz: Shifty Shapes

In Basic Mathematics, you will learn about:

• Recognizing Patterns: This is the bread and butter of being a mathematician. A quick recognition of patterns allows you to form hypothesis and ideas, paving a route to attacking the problem.
• Essentials of Angles: Picking up the terminology allows you to quickly understand what is being talked about. When in doubt, do a search for the term.
• Parallel Lines and Perpendicular Lines: Recognizing pairs of parallel and perpendicular lines allows you to exploit their properties. This is extremely useful when finding angles.
• Length and Area: Building up your familiarity with the basic shapes: triangles, rectangles, and circles will help you tremendously tackling challenging problems in geometry.