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## Introduction

Hey it's a me again @drifter1!

This is the sixth and final part of my high-school refresher series on **Geometry**. I highly suggest checking out part1, part2, part3, part4 and part5 before this part.

So, without further ado, let's get straight into it!

## Analytic Geometry Basics

A point in the **Cartesian Coordinate System**, the so called x-y plane, is denoted as **(x _{1}, y_{1})**. This doesn't represent an open interval of numbers of the form (a, b), which is commonly used in mathematics, but a point in this x-y plane. We covered the system more thoroughly in part 3. This is the basis of

**Analytical Geometry**, where mathematical equations are used in order to represent shapes and their relation.

### Delta notion

Let's start simple. Considering two points **A** and **B** in this x-y plane, moving from point **A** to **B** there will be a change in the x and/or y coordinates. This change is called a delta, **Δ** (capital Greek alphabet delta). **Δx** is the change in x, whilst **Δy** the change in y.

For example, if A = (3, 1) and B = (2, 3) then moving from A to B yields:

The general formulas for points (x_{1}, y_{1}) and (x_{2}, y_{2}) are:

### Slope

Having two distinct points A and B, only one straight line can be drawn that passes through both points. This line has a slope equal to the ratio of Δy and Δx or:

For the previous example, we have:

### Straight Line

The simplest straight line equation (the slope-intersept equation) is:

where *a* is the slope.

This equation is basically the result of refactoring the slope equation and entering any point that's on the line, such as:

This is the so called point-slope equation of a straight line.

For example, in the previous example, entering either of the two points results in the same line, which is:

The more general formula of a straight line is:

where A, B and C are constants, and A, B are not both 0.

Any *y = b* line is a horizontal line, whilst any *x = a* line is a vertical line.

### Axes Intersection Points

To find the y value for a specific x value we simply input x into the equation and solve for y (the opposite for finding x). Thus, in order to find where the line intersects the x-axis we set y = 0 and to find where the line intersects the y-axis we set x = 0. Simple stuff.

For example, for the previous line x = 0 and y = 0 yield the following intersection points:

### Parallel Lines

Two lines are parallel to each other if the slopes are equal.

For example:

is a line parallel to the previous example's line.

### Perpendicular Lines

Two lines are parallel if their slopes are negative reciprocales:

For example:

is a line perpendicular to the previous example's line as:

### Line Intersection Point

In order to find where two or more lines meet (if they meet) their linear system needs to be solved which is a topic covered by **Linear Algebra**.

For example:

### Distance Between 2 Points

The distance between two points (x_{1}, y_{1}) and (x_{2}, y_{2}) is a direct consequence of Pythagoras's Theorem and equal to:

For example, the distance between the points A and B from before is:

### Midpoints

Considering the line segment which joins two points (x_{1}, y_{1}) and (x_{2}, y_{2}), the midpoint of this segment is at:

### Conics

The simplest non-straight lines are the conics covered by the end of part 4.

Their general equation is of the form:

The Bxy term is about conic rotation. The Dx, Ey and F terms affect the vertex and center points. And lastly, for A, C we have the following shapes:

- A = C : Circle
- AC > 0 (A ≠ C) : Ellipse
- AC = 0 : Parabola
- AC < 0 : Hyperbola

For example, a circle with center at (2, -2) and radius of 2, has the following equation:

which can also be expanded into the general equation format:

These are all topics of the branch of mathematics known as **Mathematical Analysis**, which also deals with topics such as limits, differentiation, integration, sequences, series etc.

## RESOURCES:

### References

- https://www.khanacademy.org/math/geometry-home
- https://www.sfu.ca/math-coursenotes/Math%20157%20Course%20Notes/sec_AnalyticGeometry.html

### Images

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

## Final words | Next up

And this is actually it for today's post and this small refresher series!

The next "All About" articles will be about:

- Polynomial Arithmetic
- Exponentials and Logarithms
- Rational Expressions

Basically more High-School Math Refreshers!

See ya!

Keep on drifting!