The last four posts dealt with formulas for finding areas using lengths of sides, starting with the triangle, where that is all you need, and then quadrilaterals, where something more must be added; and then using coordinates of vertices. Now we can use those tools to solve some of the more common real-life problems we …
We’ve been collecting techniques for finding areas of polygons, mostly using their side lengths. We started with triangles (Heron’s formula), then quadrilaterals (Bretschneider’s formula and Brahmagupta’s formula), and the fact that the largest possible area is attained when the vertices lie on a circle. We’ll look at one more way to find area, using coordinates …
Last time we looked at applying Heron’s formula to problems about the area of a triangle, where knowing the side lengths is enough to determine the area; there was a passing mention of the fact that more is needed for quadrilaterals. We’ll start here with a repeat of that idea, and then look at several …
Last time we looked at a very useful formula for finding the area of any triangle, given only the lengths of its sides. Today I want to look at several problems in which the formula has been used, some of them surprising.
There is a beautiful formula for the area of a triangle, which many students unfortunately never get to see. In this post we’ll look at that formula and three ways to prove it; next time, I’ll show some examples of how useful it can be. (By the way, you’ll also see it called “Hero’s Formula”, …
(A new question of the week) Having just discussed some pattern or sequence problems that were poorly posed, let’s look at some recent questions about sequences, some of which are quite complicated, and others seem to be just wrong.
We’ve often pointed out that pattern or sequence problems, when nothing is given but a list of numbers, are not really math, in the sense that there is no one correct answer. They are psychology questions: What would a math teacher think is an interesting sequence to ask about? Mathematically, any number could come next, …
In the past (last May and November), we discussed ways to find patterns or sequences in numbers, sometimes leading to a formula. This included an example where the sequence turned out not to be just a provided list of numbers, but a process that generated the numbers. I want to focus on that type of …
(A new question of the week) Having just looked at L’Hôpital’s Rule, we can conclude with a look at a recent question about it, to illustrate the reality of struggling to apply it (and the process we go through to help a student find an error).
Last time we looked at the basics of L’Hôpital’s Rule, which applies to limits of the form or , and ways to understand or prove it. Here, we’ll consider a variety of questions we’ve received about less direct application of the rule. We’ll see ways to apply it to other indeterminate forms (, , ), and what …