Episode 6 of Geometry

Learn how watching a fly on his ceiling inspired the mathematician Rene Descartes to link geometry and algebra. Find out how this powerful connection allows us to use algebra to calculate distances, midpoints, and more.

Running Time

29 mins

Year

2014

Kanopy ID

1338399

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Complex Numbers in Geometry

In lecture 6, you saw how 17th-century mathematician Rene Descartes united geometry and algebra with the invention of the coordinate plane. Now go a step further and explore the power and surprises that come from using the complex number plane. Examine how using complex numbers can help solve several tricky…

Making Use of Linear Equations

Delve deeper into the connections between algebra and geometry by looking at lines and their equations. Use the three basic assumptions from previous lectures to prove that parallel lines have the same slope and to calculate the shortest distance between a point and a line.

Equidistanceâ€”A Focus on Distance

You've learned how to find the midpoint between two points. But what if you have three points? Or four points? Explore the concept of equidistance and how it reveals even more about the properties of triangles and other shapes.

Proofs and Proof Writing

The beauty of geometry is that each result logically builds on the others. Mathematicians demonstrate this chain of deduction using proofs. Learn this step-by-step process of logic and see how to construct your own proofs.

Geometry - An Interactive Journey to Mastery

Inscribed over the entrance of Plato's Academy were the words, "Let no one ignorant of geometry enter my doors." To ancient scholars, geometry was the gateway to knowledge. Its core skills of logic and reasoning are essential to success in school, work, and many other aspects of life. Yet sometimes…

The Classification of Triangles

Continue the work of classification with triangles. Find out what mathematicians mean when they use words like scalene, isosceles, equilateral, acute, right, and obtuse. Then, learn how to use the Pythagorean theorem to determine the type of triangle (even if you don't know the measurements of the angles).

The Equation of a Circle

In your study of lines, you used the combination of geometry and algebra to determine all kinds of interesting properties and characteristics. Now, you'll do the same for circles, including deriving the algebraic equation for a circle.

The Joy of Algebra Made Visual

Algebra can be used to solve geometrical problems, such as finding where two lines cross. The technique is useful in real-life problems, for example, in choosing a telephone plan. Graphs help us better understand everything from lines to equations with negative or fractional exponents.

The Pythagorean Theorem

We commonly define the Pythagorean theorem using the formula a2 + b2 = c2. But Pythagoras himself would have been confused by that. Explore how this famous theorem can be explained using common geometric shapes (no fancy algebra required), and how it's a critical foundation for the rest of geometry.

Similarity and Congruence

Define what it means for polygons to be "similar" or "congruent" by thinking about photocopies. Then use that to prove the third key assumption of geometry--the side-angle-side postulate--which lets you verify when triangles are similar. Thales of Ionia used this principle in 600 B.C.E. to impress the Egyptians by calculating…

A Return to Parallelism

Continue your study of parallelism by exploring the properties of transversals (lines that intersect two other lines). Prove how corresponding angles are congruent, and see how this fact ties into a particular type of polygon: trapezoids.

I Want To Be A Mathematician: A Conversation with Paul Halmos

A 44-minute interview with mathematician Paul Halmos that touches on the Moore Method, becoming a mathematician, great teachers, designing a course, writing, and the state of education in the United States. The interview conducted in 1999 by Peter Renz and George Csicsery was released by the Mathematical Association of America…

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