Is a Circle One-Dimensional or Two-Dimensional?
A recent question led me to find a series of old questions, each answering it at a different level. All of them are worth presenting.
Geometry
Internal and External Division of a Segment
Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …
Dividing a Segment: Get the Order Right!
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …
Trouble with Transformations
Having looked into our explanations of transformations and symmetry, over the last weeks, let’s turn to the recent questions that triggered this series. Here we have an adult studying the topic from a good book, but tripping over some issues in the book. We’ll be touching on some topics we haven’t yet looked at, such …
From Transformations to Symmetry
Having looked at geometrical transformations, we can now apply them to the idea of symmetry. We’ll focus on symmetry of figures in a plane.
Slides, Turns, and Flips: How to Combine Them
Last time we looked at what it means to translate, rotate, and reflect figures on a plane. Here, we’ll look at some questions about what happens when these three transformations (and a fourth, the glide reflection) are combined.
Slides, Turns, and Flips: Transformations in Geometry
Some recent questions have dealt with translation, rotation, and reflection of geometric shapes, so it may be time to look into what we have said about that in the past. Here we’ll look at the meaning of these terms, at levels suitable for both kids and adults, and next time we’ll see what happens when …
Slides, Turns, and Flips: Transformations in Geometry Read More »
Ratios and Areas: An Unusual Pie Chart Problem
Here is a short problem with several levels of difficulty. The problem itself is poorly designed, as we’ll see, but still provides several useful lessons, dealing with measurement, rounding, and ratios.
How to Find Any Part of a Segment of a Circle
Recent questions have dealt with calculations of various parts of a segment of a circle (chord, arc, sagitta, etc.). How, for example, can you find the arc length if you know the chord length and the height? The definitive explanation of these questions is found in a classic page from Ask Dr. Math, written by …
Finding Length of a Roll: Facing Reality
Last time we looked at how to find the length of material on a roll, making some necessary simplifications. Here, I want to look at some variations on that: first, about carpet in particular, and then about wire on a spool.