Introduction to Transformations and Coordinate Geometry Review
TL;DR
Transformations move points and shapes on a plane, and coordinate geometry provides the language to describe these movements precisely using numbers. You'll review how points, lines, and shapes are represented numerically and how to calculate essential properties like distance and midpoint. This foundational knowledge is crucial for understanding how transformations change geometric objects.
1. The Mental Model
Think of transformations as actions like sliding, turning, or flipping a picture on a table. Coordinate geometry is like the grid system on the table that tells you exactly where everything is before and after those actions. It allows you to describe these movements mathematically.
2. The Core Material
Representing Points
A point on a 2D plane is uniquely identified by an ordered pair of numbers, $(x, y)$, where $x$ is the horizontal position (left/right of the origin) and $y$ is the vertical position (up/down from the origin). The origin is the point $(0,0)$.
Representing Lines
A line is a continuous set of points. You can describe a line in a few ways:
* Two points: A line is uniquely defined by any two distinct points it passes through, say $(x_1, y_1)$ and $(x_2, y_2)$.
* Equation: The most common form is the slope-intercept form: $y = mx + c$, where $m$ is the gradient (slope) and $c$ is the y-intercept (where the line crosses the y-axis).
* Gradient ($m$): Measures the steepness of the line. $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$.
* General form: $Ax + By + C = 0$.
Representing Shapes
A shape (like a triangle or a square) is defined by its vertices (corner points). For example, a triangle is defined by three points: $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$.
Distance Between Two Points
To find the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the distance formula, which comes from the Pythagorean theorem:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint of a Line Segment
The midpoint $M$ of a line segment connecting two points $(x_1, y_1)$ and $(x_2, y_2)$ is found by averaging their x-coordinates and y-coordinates:
$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
3. Worked Example
Let's say you have two points, $A = (1, 2)$ and $B = (5, 6)$.
- Calculate the distance between A and B.
Using