Welcome to Coordinate Geometry!
In this chapter, we are going to explore Straight Lines. You’ve been looking at graphs for years, but now we’re going to master the "language" of lines. Think of coordinate geometry as a bridge between algebra and shapes. It allows us to use equations to describe exactly where things are and how they move.
Why does this matter? Architects use these equations to design buildings, programmers use them to create hit video games, and economists use them to predict market trends. Don't worry if you find some of the algebra a bit "stretchy" at first—we’ll break it down step-by-step!
1. The Foundations: Midpoints and Distance
Before we build the lines, we need to know how to measure the gaps between points.
The Midpoint: Meeting Halfway
Imagine you and a friend live at different coordinates on a map and want to meet exactly halfway for a coffee. To find that middle spot, you just find the average of your coordinates.
If you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the midpoint is:
\( \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Quick Hint: Just remember: "Add them up and divide by 2!"
The Distance Between Two Points
To find the distance between two points, we use a formula that is actually just Pythagoras’ Theorem in disguise. If you draw a right-angled triangle between two points, the distance is the hypotenuse.
The distance \(d\) between \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Did you know? Because we square the differences, it doesn't matter if you do \( (x_2 - x_1) \) or \( (x_1 - x_2) \). The result will always be positive!
Key Takeaway: Use averages for the middle, and use Pythagoras for the length.
2. The Gradient (Steepness)
The gradient (usually called \(m\)) tells us how steep a line is. It is the "Rise over Run."
\( m = \frac{y_2 - y_1}{x_2 - x_1} \)
- If \(m\) is positive, the line goes up as you move right.
- If \(m\) is negative, the line goes down as you move right.
- If \(m = 0\), the line is horizontal.
Common Mistake to Avoid: Always make sure you subtract the coordinates in the same order. If you start with \(y_2\) on top, you must start with \(x_2\) on the bottom!
3. Three Ways to Write a Line Equation
The OCR syllabus requires you to be comfortable with three different "outfits" a straight-line equation can wear.
Form 1: The Slope-Intercept Form
\( y = mx + c \)
This is the one you likely know best. \(m\) is the gradient and \(c\) is the y-intercept (where the line crosses the vertical axis).
Form 2: The Point-Gradient Form
\( y - y_1 = m(x - x_1) \)
Struggling student tip: This is often the most useful form! If you have the gradient \(m\) and any point \( (x_1, y_1) \) on the line, you just plug them in. You don't need to go hunting for the y-intercept first.
Form 3: The General Form
\( ax + by + c = 0 \)
In this form, \(a\), \(b\), and \(c\) are usually integers (whole numbers). This is a very neat way to write equations and is often requested in exam questions.
Step-by-Step: Finding a line through two points
1. Find the gradient \(m\) using the gradient formula.
2. Pick one of the points to be your \( (x_1, y_1) \).
3. Use the formula \( y - y_1 = m(x - x_1) \).
4. Rearrange into the form the question asks for (like \(y = mx + c\)).
Key Takeaway: \( y - y_1 = m(x - x_1) \) is your best friend for building equations quickly.
4. Parallel and Perpendicular Lines
How do lines relate to each other? Their gradients tell the whole story.
Parallel Lines
Parallel lines are like train tracks—they never meet because they have the same steepness.
Condition: \( m_1 = m_2 \)
Perpendicular Lines
Perpendicular lines meet at a perfect right angle (\(90^\circ\)). Their gradients are negative reciprocals of each other.
Condition: \( m_1 \times m_2 = -1 \)
Example: If a line has a gradient of \(3\), a line perpendicular to it will have a gradient of \( -\frac{1}{3} \).
If a line has a gradient of \( -\frac{2}{5} \), the perpendicular gradient is \( \frac{5}{2} \).
Memory Trick: To find a perpendicular gradient, "Flip it and Switch it" (Flip the fraction upside down and switch the plus/minus sign).
Key Takeaway: Parallel = Same gradient. Perpendicular = Gradients multiply to make \(-1\).
5. Points of Intersection
When two lines cross, they share a single coordinate \( (x, y) \). To find this point, you simply solve the two equations simultaneously.
- Substitution Method: If one equation is \( y = ... \), plug that "..." into the \(y\) of the other equation.
- Elimination Method: Line up the equations and add or subtract them to get rid of \(x\) or \(y\).
Don't worry if this feels like a lot of algebra! It’s just finding the one pair of numbers that works for both lines at the same time.
6. Real-World Modeling
Straight lines aren't just for grids; they model rates of change. If a plumber charges a flat fee of £40 plus £20 per hour, we can model this as:
\( y = 20x + 40 \)
Where \(y\) is the total cost and \(x\) is the number of hours. The gradient is the hourly rate, and the y-intercept is the fixed starting fee.
Quick Review Box:
- Midpoint: Average of \(x\)'s, average of \(y\)'s.
- Distance: \( \sqrt{\text{change in } x^2 + \text{change in } y^2} \).
- Gradient: \( \frac{y_2 - y_1}{x_2 - x_1} \).
- Perpendicular: Flip the fraction and change the sign.
- Equation: \( y - y_1 = m(x - x_1) \).
Summary Checklist
Before moving on to Circles, make sure you can:
1. Calculate the distance and midpoint between two points.
2. Find the gradient of a line.
3. Write the equation of a line using any of the three standard forms.
4. Find the equation of a line parallel or perpendicular to another.
5. Find where two lines cross by solving equations.