Hello and Welcome to Coordinate Geometry!
This chapter, Length and Midpoint, is essential for understanding how to work with coordinates in two dimensions. Think of coordinate geometry as using an imaginary map (the Cartesian plane) to calculate real-world values.
In these notes, we will learn two fundamental skills:
- Calculating the distance (length) between two points.
- Finding the exact centre (midpoint) between two points.
These concepts are crucial for solving complex geometry problems later on, so let's master them!
Section 1: Calculating the Length of a Line Segment
The Distance Formula: Finding 'As the Crow Flies'
When you have two points, \(A\) and \(B\), on a graph, the length of the line segment connecting them is the shortest distance between them. How do we find this length using only the coordinates? We use the brilliant idea behind Pythagoras' Theorem.
Prerequisite: The Right-Angled Triangle Connection
Imagine you are traveling from point \(A\) to point \(B\). Instead of taking the direct diagonal line (which is the length we want), you can only move horizontally (change in \(x\)) and vertically (change in \(y\)).
- The horizontal path is the length of the triangle's base.
- The vertical path is the height of the triangle.
- The line segment length is the hypotenuse (the longest side).
Since we have a right-angled triangle, we know that:
\((\text{Length})^2 = (\text{Horizontal Distance})^2 + (\text{Vertical Distance})^2\)
The Official Distance Formula (Extended Content E4.3, Core Content C4.3)
Let the two points be \(A(x_1, y_1)\) and \(B(x_2, y_2)\).
The length, \(D\), is calculated using the Distance Formula:
\[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Step-by-Step Guide for Finding Length
- Label the Points: Decide which point is \((x_1, y_1)\) and which is \((x_2, y_2)\). (It doesn't matter which one you pick first!)
- Find the Horizontal Change (\(\Delta x\)): Calculate the difference between the \(x\)-coordinates: \((x_2 - x_1)\).
- Find the Vertical Change (\(\Delta y\)): Calculate the difference between the \(y\)-coordinates: \((y_2 - y_1)\).
- Square the Differences: Square both results from Steps 2 and 3. (This ensures the result is always positive, which is vital since length must be positive.)
- Add and Square Root: Add the squared results together, and finally, take the square root of the total sum.
Example: Find the length of the line segment joining \(P(-2, 3)\) and \(Q(4, -5)\).
1. Label: \(x_1=-2, y_1=3\) and \(x_2=4, y_2=-5\).
2. Horizontal Change: \((4 - (-2)) = 6\)
3. Vertical Change: \((-5 - 3) = -8\)
4. Square: \(6^2 = 36\) and \((-8)^2 = 64\)
5. Add and Square Root: \(D = \sqrt{36 + 64} = \sqrt{100} = 10\) units.
🧠 Common Mistakes to Avoid (Distance)
- Forgetting the Square Root: The formula gives you \(D^2\) before the final step. Don't forget to take the square root at the end!
- Mistake with Negatives: Remember that squaring a negative number always results in a positive number (e.g., \((-8)^2 = 64\)). If you get a negative number under the square root, you have made an error.
Key Takeaway for Length
The distance formula is just Pythagoras' Theorem dressed up in coordinates. It measures the hypotenuse of the triangle formed by the horizontal and vertical distances.
Section 2: Finding the Midpoint of a Line Segment
The Midpoint Formula: Finding the Exact Middle
The midpoint is the exact center point of a line segment. Since it is a point, the answer must be given as a pair of coordinates \((x, y)\).
Analogy: Finding the Average Position
If you are meeting a friend who lives at position 10 on a line, and you live at position 50, where is the middle ground? It's the average: \((10 + 50) / 2 = 30\).
Finding the midpoint in coordinate geometry is exactly the same, but you do it separately for the \(x\)-coordinates and the \(y\)-coordinates.
The Official Midpoint Formula (Extended Content E4.3, Core Content C4.3)
Let the two points be \(A(x_1, y_1)\) and \(B(x_2, y_2)\).
The midpoint, \(M\), has coordinates:
\[M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]
It is the average of the \(x\)'s, and the average of the \(y\)'s.
Step-by-Step Guide for Finding Midpoint
- Label the Points: Assign \((x_1, y_1)\) and \((x_2, y_2)\).
- Find the Mid-x: Add the \(x\)-coordinates together, then divide the sum by 2.
- Find the Mid-y: Add the \(y\)-coordinates together, then divide the sum by 2.
- Write the Result: Present the answer as a coordinate pair \((x_{\text{mid}}, y_{\text{mid}})\).
Example: Find the midpoint of the line segment joining \(R(5, 1)\) and \(S(-1, 9)\).
1. Label: \(x_1=5, y_1=1\) and \(x_2=-1, y_2=9\).
2. Mid-x: \(\frac{5 + (-1)}{2} = \frac{4}{2} = 2\)
3. Mid-y: \(\frac{1 + 9}{2} = \frac{10}{2} = 5\)
4. Result: The midpoint \(M\) is \((2, 5)\).
Did you know? (Connection to Vectors)
Finding the midpoint is closely related to vectors. If you translate from one point to the other, the midpoint is achieved by translating half the distance along that vector!
🧠 Common Mistakes to Avoid (Midpoint)
- Mixing up Addition and Subtraction: Midpoint requires adding the coordinates. Distance requires subtracting them. Don't confuse the two!
- Forgetting to Divide: You must divide the sum of the coordinates by 2 to find the average.
- Incorrect Format: The final answer must be a coordinate \((x, y)\), not just two separate numbers.
Quick Review Box: Length vs. Midpoint
To help struggling students keep these formulas straight, remember the action words:
| Concept | What do you do? | Key Operation | Formula Hint |
|---|---|---|---|
| Length (Distance) | Find the change (difference) and use Pythagoras. | Subtract and Square Root. | \(D = \sqrt{(\text{Diff } x)^2 + (\text{Diff } y)^2}\) |
| Midpoint | Find the average position. | Add and Divide by 2. | \(M = (\frac{\text{Sum } x}{2}, \frac{\text{Sum } y}{2})\) |
Summary: Key Takeaways
You now have the tools to precisely locate the center of any line segment and measure its exact length on the Cartesian plane. Practice substituting coordinates carefully, especially when dealing with negative numbers. You've got this!