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 $B = (5, 6)$.

1. Calculate the distance between A and B.
   Using the distance formula:
   $d = \sqrt{(5 - 1)^2 + (6 - 2)^2}$
   $d = \sqrt{(4)^2 + (4)^2}$
   $d = \sqrt{16 + 16}$
   $d = \sqrt{32}$
   $d = 4\sqrt{2}$ (approximately 5.66 units)

2. Find the midpoint of the line segment AB.
   Using the midpoint formula:
   $M = \left( \frac{1 + 5}{2}, \frac{2 + 6}{2} \right)$
   $M = \left( \frac{6}{2}, \frac{8}{2} \right)$
   $M = (3, 4)$

3. Find the gradient of the line passing through A and B.
   Using the gradient formula:
   $m = \frac{6 - 2}{5 - 1}$
   $m = \frac{4}{4}$
   $m = 1$

4. Write the equation of the line passing through A and B in slope-intercept form.
   You know $m=1$. Now use one of the points (e.g., A=(1,2)) to find $c$ in $y=mx+c$:
   $2 = (1)(1) + c$
   $2 = 1 + c$
   $c = 1$
   So, the equation of the line is $y = x + 1$.

4. Key Takeaways

• Points are fundamental, described by $(x,y)$ coordinates on a 2D plane.
• Lines can be defined by two points, or by an equation like $y=mx+c$.
• The gradient ($m$) tells you how steep a line is.
• Shapes are defined by their vertices, which are just a collection of points.
• The distance formula helps you find the length of a line segment.
• The midpoint formula helps you find the exact centre of a line segment.
• Coordinate geometry provides the numeric backbone for all geometric transformations.

Common mistakes to avoid:
- Mixing up $x_1/y_1$ and $x_2/y_2$ consistently within a formula (e.g., $(x_1-x_2)$ in one part, but $(y_2-y_1)$ in another).
- Forgetting the square root in the distance formula.
- Incorrectly calculating the slope as $\frac{\Delta x}{\Delta y}$ instead of $\frac{\Delta y}{\Delta x}$.
- Forgetting that the general form $Ax+By+C=0$ needs to be set to zero.

5. Now Try It

Given three points $P=(-2, 3)$, $Q=(4, 5)$, and $R=(0, -1)$, calculate the lengths of the line segments PQ, QR, and RP. Then, find the midpoints of PQ and PR. Finally, determine the gradient and the equation (in slope-intercept form) of the line passing through points P and Q. What does the sum of the lengths of PQ, QR, and RP tell you? (Hint: it's the perimeter of a triangle).