4.5. Geometry: Vectors, Curves…Formulas From Your Past?#
4.5.1. Curves#
4.5.1.1. Equation of a circle#
We will deal with some functions that would be very hard to evaluate on your calculator. But Descartes’ gift is that I can show you the graph and evaluation can be done by eye, which is in effect solving the equation. We’ll use some simple geometrical relations which I’ll summarize here.
A circle of radius \(R\) in the \(x-y\) plane centered at a \((a,b)\) is described by the equation:
Of course if the circle is centered at the origin, then it looks more familiar as in this figure:
4.5.1.2. Equation of a parabola#
A parabola in the \(x-y\) plane facing up with vertex at \((a,b)\) where \(C\) is a constant has the equation,
4.5.1.3. Area of a rectangle#
A rectangle with sides \(a\) and \(b\) has an area, \(A\) of
4.5.1.4. Area of a right triangle#
A right triangle (which means that one of the angles is \(90\) degrees) with base of \(a\) and height of \(b\) has an area, \(A\) of
For a right triangle, the base and height are equal to the two legs. But the formula works for any triangle. Here are some examples,
4.5.1.5. Area and circumference of a circle#
Circles will involve \(\pi\) which is an irrational number for which we’ll never need precision better than “March 14th,” \(\pi = 3.14\). And, yes, the story is true that in 1897 Indiana’s General Assembly tried to change the value to \(\pi_{\text{Indiana}}=3.0\) in order to simplfy calculations. Mathematicians and engineers around the state had collective heart palpitations. It disappeared quickly.
For a circle of radius \(R\), the area, \(A\) is
and the circumference, \(C\) is
4.5.2. Pythagoras’ Theorem#
For a right triangle (like the left hand triangle above), the hypotenuse, \(c\) is related to the lengths of the two sides \(a\) and \(b\) by the Theorem of Pythagoras:
And, no. He didn’t invent it and it’s been proven many, many different ways.
4.5.3. The quadratic formula#
We might run across a particular polynomial, which you’ve also probably seen before:
It’s an “order 2” polynomial, which means that there are two values of \(x\) that qualify as “solutions”: the values of \(x\) that when substituted make the function be zero. You could plot the function and find what \(x\) values the curve passes through the \(x-\)axis, or you could rely on the time-honored recipe:
