The topic in my geometry class today involved finding solutions to quadratic equations. The actual topic was on the ratio of lengths of secants in a circle, but that’s for another post. For a specific example, consider this problem. Two secants intersect outside a circle. The first secant line has segments of 15 (between the…
Month: May 2014
There are things
There are things I wish I could say where you could hear them but the smile I painted on to the rhythm of the metronome cannot be so easily belied as that Somewhere at night my faith capsized, rammed against a frozen memory floating on the surface And for a moment, I reached out and…
Pascal’s Triangle and Dice Rolls
Pascal’s Triangle Pascal’s Triangle represents the coefficients of a binomial such as \(x + 1\) raised to a power. Row n of the triangle lists the coefficients of \((x + 1)^{n-1}\). Here are the first few rows of Pascal’s Triangle: \[\newcommand\cn[3]{\llap{#1}#2\rlap{#3}} \begin{array}{c} &&&&&&\cn{}{1}{}\\ &&&&&\cn{}{1}{}&&\cn{}{1}{}\\ &&&&\cn{}{1}{}&&\cn{}{2}{}&&\cn{}{1}{}\\ &&&\cn{}{1}{}&&\cn{}{3}{}&&\cn{}{3}{}&&\cn{}{1}{}\\ &&\cn{}{1}{}&&\cn{}{4}{}&&\cn{}{6}{}&&\cn{}{4}{}&&\cn{}{1}{}\\ &\cn{}{1}{}&&\cn{}{5}{}&&\cn{1}{}{0}&&\cn{1}{}{0}&&\cn{}{5}{}&&\cn{}{1}{}\\ \cn{}{1}{}&&\cn{}{6}{}&&\cn{1}{}{5}&&\cn{2}{}{0}&&\cn{1}{}{5}&&\cn{}{6}{}&&\cn{}{1}{} \end{array}\] For instance, row 4 is…
Solving Simultaneous Equations: Multiple Methods
Introduction and Terms Recently, a post on the G+ Mathematics community involved how to determine \(x\) and \(y\) when: \[3x + 5y = 12 \\ x + y = 2\] This is generally referred to as simultaneous equations or a system of equations. As a general rule, for such a problem to be solvable, you…
Finding an Incenter via Formulas
Terms For every polygon, there is a largest circle that fits entirely within that polygon. If that circle touches all sides of the polygon, then it is said to be inscribed; it is called the incircle, and its center is called the incenter (which is then also called the polygon’s incenter). Every triangle has an incenter…