Introduction to Coordinates: Your Guide to Mapping the Maths World!
Hello there! Welcome to the amazing world of coordinates. Ever played a video game where you have to find a character on a map? Or used a map app to find your way? That's all thanks to coordinates!
This chapter is all about learning how to describe the exact location of a point on a flat surface, called a plane. It's like learning the secret code to a treasure map. By the end of this, you'll be able to plot points, find distances, and even move shapes around, all using numbers. Don't worry if it sounds tricky at first – we'll break it down step-by-step. Let's begin our adventure!
1. The Rectangular Coordinate System: Our Map
To find locations, we need a map. In maths, our map is called the Rectangular Coordinate System or the Cartesian Plane. It looks like a grid with two main lines crossing each other.
The Key Parts of Our Map
- The x-axis: This is the horizontal line that goes left and right. Think of it as a corridor you walk along.
- The y-axis: This is the vertical line that goes up and down. Think of it as the stairs you climb up or down.
- The Origin: This is the special point where the x-axis and y-axis cross. It's our starting point, and its address is always (0, 0).
Coordinates: The Address of a Point
Every point on the plane has a unique address called its coordinates, which we write as an ordered pair: (x, y).
- The first number is the x-coordinate. It tells you how far to move left or right from the origin.
- The second number is the y-coordinate. It tells you how far to move up or down.
Memory Aid: Run Before You Jump!
A simple way to remember the order (x, y) is: You have to 'run' (along the horizontal x-axis) before you can 'jump' (up or down the vertical y-axis). Also, X comes before Y in the alphabet!
Example: Let's find the point P(3, 2).
1. Start at the Origin (0, 0).
2. 'Run' 3 units to the right along the x-axis (because the x-coordinate is positive 3).
3. From there, 'Jump' 2 units up (because the y-coordinate is positive 2).
4. Mark that spot. That's point P!
The Four Quadrants
The x and y axes divide the plane into four sections called quadrants. The signs of the coordinates tell you which quadrant a point is in.
- Quadrant I: Top right. Here, both x and y are positive. (+, +)
- Quadrant II: Top left. Here, x is negative and y is positive. (-, +)
- Quadrant III: Bottom left. Here, both x and y are negative. (-, -)
- Quadrant IV: Bottom right. Here, x is positive and y is negative. (+, -)
Quick Review: The Basics
Coordinate Plane: Our grid-like map.
x-axis: The horizontal line.
y-axis: The vertical line.
Origin: The starting point (0, 0).
Coordinates: The address of a point, written as (x, y).
Key Takeaway
The coordinate system gives every point a unique (x, y) address, telling us exactly where it is on the plane.
2. Finding Simple Distances
Now that we can plot points, let's find the distance between them. We'll start with the easy ones: points on a straight horizontal or vertical line.
Distance Between Two Points on a Horizontal Line
If two points are on a horizontal line, their y-coordinates will be the same. To find the distance, just find the difference between their x-coordinates.
Formula: Distance = $$|x_2 - x_1|$$ (The vertical bars mean absolute value, which just means the positive difference).
Example: Find the distance between A(2, 3) and B(7, 3).
1. Notice that the y-coordinates are both 3, so it's a horizontal line.
2. Subtract the x-coordinates: 7 - 2 = 5.
3. The distance between A and B is 5 units.
Distance Between Two Points on a Vertical Line
If two points are on a vertical line, their x-coordinates will be the same. To find the distance, just find the difference between their y-coordinates.
Formula: Distance = $$|y_2 - y_1|$$
Example: Find the distance between C(4, 1) and D(4, 6).
1. Notice that the x-coordinates are both 4, so it's a vertical line.
2. Subtract the y-coordinates: 6 - 1 = 5.
3. The distance between C and D is 5 units.
Common Mistake to Avoid!
Make sure you subtract the correct coordinates! For horizontal lines, use the x-values. For vertical lines, use the y-values. A quick sketch on a grid can help you see which one to use.
Key Takeaway
For points on a horizontal or vertical line, the distance is simply the positive difference between the coordinates that are not the same.
3. The Distance Formula (for any two points!)
What if the line isn't perfectly horizontal or vertical? How do we find the distance then? We use a powerful tool called the Distance Formula. It works for ANY two points.
Did you know? The distance formula is really just the Pythagoras' Theorem in disguise! It calculates the length of the hypotenuse of a right-angled triangle formed by the two points.
The Formula
For any two points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$, the distance d between them is:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$Step-by-Step Guide to Using the Formula
Example: Find the distance between P(1, 2) and Q(4, 6).
Step 1: Label your coordinates.
Let P be the first point: $$x_1 = 1, y_1 = 2$$.
Let Q be the second point: $$x_2 = 4, y_2 = 6$$.
(It doesn't matter which point you call the first or second, you'll get the same answer!)
Step 2: Plug the values into the formula.
$$d = \sqrt{(4 - 1)^2 + (6 - 2)^2}$$
Step 3: Solve inside the brackets first.
$$d = \sqrt{(3)^2 + (4)^2}$$
Step 4: Square the numbers.
$$d = \sqrt{9 + 16}$$
Step 5: Add the numbers under the square root.
$$d = \sqrt{25}$$
Step 6: Find the square root.
$$d = 5$$
The distance between P and Q is 5 units.
Key Takeaway
The distance formula helps us find the length of a straight line between any two points on the coordinate plane.
4. The Mid-point Formula: Finding the Middle
The mid-point is the point that is exactly halfway between two other points. Think of it as finding the average location. To find it, we just find the average of the x-coordinates and the average of the y-coordinates.
The Formula
For any two points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$, the mid-point M is:
$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$Step-by-Step Guide to Using the Formula
Example: Find the mid-point between A(2, 3) and B(8, 7).
Step 1: Identify your x's and y's.
$$x_1 = 2, x_2 = 8$$
$$y_1 = 3, y_2 = 7$$
Step 2: Find the average of the x-coordinates.
$$\frac{2 + 8}{2} = \frac{10}{2} = 5$$
Step 3: Find the average of the y-coordinates.
$$\frac{3 + 7}{2} = \frac{10}{2} = 5$$
Step 4: Write the new coordinates.
The mid-point is M(5, 5).
Key Takeaway
The mid-point formula gives you the coordinates of the halfway point between two other points by averaging their x and y values.
5. Slope of a Line: How Steep Is It?
The slope (often shown with the letter m) is a number that tells us how steep a line is. It's the "rise over run".
- Rise: The vertical change between two points (how much it goes up or down).
- Run: The horizontal change between two points (how much it goes left or right).
The Formula
For any two points $$A(x_1, y_1)$$ and $$B(x_2, y_2)$$, the slope m is:
$$m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}$$Four Types of Slope
- Positive Slope: The line goes uphill from left to right.
- Negative Slope: The line goes downhill from left to right.
- Zero Slope: The line is perfectly horizontal. (Rise is 0).
- Undefined Slope: The line is perfectly vertical. (Run is 0, and we can't divide by zero!).
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 1}{4 - 2} = \frac{4}{2} = 2$$
The slope is 2. This is a positive slope, so the line goes uphill.
Key Takeaway
Slope (m) measures the steepness of a line. A bigger number means a steeper line. The sign tells you if it goes uphill (+) or downhill (-).
6. Parallel and Perpendicular Lines
The slope of a line can also tell us about its relationship with another line.
Parallel Lines
Parallel lines are lines that never, ever cross. Think of train tracks. For this to happen, they must have the exact same steepness.
Rule: Two lines are parallel if they have the same slope.
$$m_1 = m_2$$
Example: A line with slope 3 is parallel to another line with slope 3.Perpendicular Lines
Perpendicular lines are lines that cross at a perfect right angle (90°). Their slopes have a special relationship.
Rule: Two lines are perpendicular if their slopes are negative reciprocals. This means if you multiply their slopes, you get -1.
$$m_1 \times m_2 = -1$$
What is a negative reciprocal? Just flip the fraction and change the sign!
Example: If a line has a slope of $$m_1 = 2$$, the slope of a perpendicular line would be $$m_2 = -\frac{1}{2}$$.(Check: $$2 \times -\frac{1}{2} = -1$$)
Key Takeaway
Slopes tell us if lines are parallel (same slope) or perpendicular (slopes multiply to -1).
7. Transformations: Moving Points Around
A transformation is a way of moving or changing a point or a shape on the coordinate plane. Let's look at the three main types: translation, reflection, and rotation.
Translation (Sliding)
This is the easiest one! It just means "sliding" a point to a new location without turning or flipping it.
- To move a point right by 'a' units, add 'a' to the x-coordinate: $$(x, y) \rightarrow (x+a, y)$$
- To move a point left by 'a' units, subtract 'a' from the x-coordinate: $$(x, y) \rightarrow (x-a, y)$$
- To move a point up by 'b' units, add 'b' to the y-coordinate: $$(x, y) \rightarrow (x, y+b)$$
- To move a point down by 'b' units, subtract 'b' from the y-coordinate: $$(x, y) \rightarrow (x, y-b)$$
New x = 3 - 2 = 1
New y = 4 - 5 = -1
The new point is A'(1, -1). (We often use an apostrophe ' to show the new point).
Reflection (Flipping)
This is like creating a mirror image of a point across a "line of reflection".
- Reflecting across the x-axis: The x-coordinate stays the same, the y-coordinate changes sign.
$$(x, y) \rightarrow (x, -y)$$ - Reflecting across the y-axis: The y-coordinate stays the same, the x-coordinate changes sign.
$$(x, y) \rightarrow (-x, y)$$
The new point is B'(5, -2).
Example: Reflect the point C(-3, 4) across the y-axis.The new point is C'(3, 4).
Rotation about the Origin (Turning)
This means turning a point around the origin (0, 0) by a certain angle. The rules below are for anti-clockwise rotation.
- Rotation by 90°: Swap the x and y coordinates, and change the sign of the new x-coordinate.
$$(x, y) \rightarrow (-y, x)$$ - Rotation by 180°: Just change the signs of both coordinates.
$$(x, y) \rightarrow (-x, -y)$$ - Rotation by 270°: Swap the x and y coordinates, and change the sign of the new y-coordinate.
$$(x, y) \rightarrow (y, -x)$$
A 90° clockwise rotation is the same as a 270° anti-clockwise rotation!
Example: Rotate the point P(2, 3) by 90° anti-clockwise about the origin.Using the rule (x, y) → (-y, x), the new point is P'(-3, 2).
Key Takeaway
Transformations move points to new locations. Translation slides, reflection flips, and rotation turns. Each one has a specific rule for changing the coordinates.
8. Finding Areas of Polygons
We can use our knowledge of coordinates and distances to find the area of simple shapes like rectangles and triangles.
Strategy: Find the Base and Height
The best way to do this is to plot the points and find the lengths of the sides you need (like base and height) using the simple distance rules for horizontal and vertical lines.
Example: Find the area of a rectangle with vertices at A(1, 1), B(7, 1), C(7, 4), and D(1, 4).
Step 1: Find the length of the base.
The base can be the line segment AB. This is a horizontal line.
Length of AB = $$|7 - 1| = 6$$ units.
Step 2: Find the length of the height.
The height can be the line segment BC. This is a vertical line.
Length of BC = $$|4 - 1| = 3$$ units.
Step 3: Calculate the area.
Area of a rectangle = Base × Height
Area = 6 × 3 = 18 square units.
Key Takeaway
By plotting the vertices of a polygon, you can find the lengths of its horizontal and vertical sides and use them to calculate its area.