## Geometry: Vectors, Curves…Formulas From Your Past?

### Curves 

#### 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:

$$(x-a)^2 + (y-b)^2 = R^2.$$

Of course if the circle is centered at the origin, then it looks more familiar as in this figure:

```{figure} ./../_images/beginning/circle.png
---
width: 550px
name: circle
align: center
---
A circle centered at the origin described by the equation, $x^2 + y^2 = 81$. It has radius of $9$, area $A= \\pi 9^2$, and circumference $C=2 \\pi 9$
```

#### 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,

$$y=C(x-a)^2 + b.$$

```{figure} ./../_images/beginning/parabola.png
---
width: 550px
name: parabola
align: center
---
A parabola satisfying the equation, $y = 1x^2$.
```


#### Area of a rectangle

A rectangle with sides $a$ and $b$ has an area, $A$ of

$$A = ab$$

#### 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

$$A= \frac{1}{2} ab.$$

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,

```{figure} ./../_images/beginning/threetriangles.png
---
width: 550px
name: threetriangles
align: center
---
Three triangles, all with the same areas.
```


#### 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.

```{figure} ./../_images/beginning/pizza.png
---
width: 350px
name: pizza
align: center
---
You realize that two pizzas is a 'circumference'? Because...wait for it...it’s '2 pie are.' You’re welcome.
```

For a circle of radius $R$, the area, $A$ is

$$A=\pi R^2$$

and the circumference, $C$ is

$$C = 2\pi R$$

### 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:

$$c^2 = a^2 + b^2.$$

And, no. He didn't invent it and it's been proven many, many different ways.

### The quadratic formula

We might run across a particular polynomial, which you've also probably seen before:

$$f(x) = ax^2 + bx + c. \nonumber$$

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:

$$x_{1,2} = \frac{-b \pm \sqrt{b^2 -4ac}}{2a} \nonumber$$

```{figure} ./../_images/beginning/stop.png
---
width: 350px
name: stop
align: center
---
You can stop here. The rest is for reference.
```