Mathematics (0580) | Length and midpoint

Coordinate Geometry: Length and Midpoint Study Notes

Hey mathematicians! Welcome to the exciting world of Coordinate Geometry. This chapter is all about finding specific information—like how far apart two points are, and exactly where the middle of a line lies—when those points are placed on a graph.

Think of it like being an air traffic controller: you need to know the exact distance between two planes (length) and the perfect halfway point for refueling (midpoint). Mastering these two simple formulas gives you serious power!

Section 1: The Distance Formula (Length of a Line Segment)

When you have two points, say Point A and Point B, calculating the straight-line distance between them requires a special tool: the Distance Formula. This formula is actually just a sneaky application of Pythagoras' Theorem (\(a^2 + b^2 = c^2\))!

1.1 The Concept: Pythagoras on the Grid

If you plot two points, \((x_1, y_1)\) and \((x_2, y_2)\), you can always draw a right-angled triangle connecting them.

• The horizontal side (\(a\)) is the difference in the x-coordinates.
• The vertical side (\(b\)) is the difference in the y-coordinates.
• The distance (\(c\)) is the hypotenuse (the line segment itself).

1.2 The Official Distance Formula

The length (\(L\)) of the line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

$$L = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

(Don't worry if this formula looks intense—we break it down step-by-step!)

Step-by-Step Guide for Length Calculation

1. Label Your Points: Clearly assign \((x_1, y_1)\) to your first point and \((x_2, y_2)\) to your second point. (It doesn't matter which point is which, since we square the result!).
2. Find the Change in x (\(\Delta x\)): Calculate the difference: \((x_2 - x_1)\).
3. Find the Change in y (\(\Delta y\)): Calculate the difference: \((y_2 - y_1)\).
4. Square the Differences: Square the result from Step 2 and Step 3. (This ensures both terms are positive).
5. Add the Squares: Add the results from Step 4 together.
6. Take the Square Root: Find the square root of the final sum to get the length \(L\).

Memory Trick: Think of the sequence: Subtract, Square, Add, Root! (SSAR)

Example Calculation (Length)

Find the length of the line segment between A (2, 5) and B (8, 13).

1. Label: \(x_1 = 2, y_1 = 5\) and \(x_2 = 8, y_2 = 13\)

2. Change in x: \((8 - 2) = 6\)

3. Change in y: \((13 - 5) = 8\)

4. Square them: \(6^2 = 36\) and \(8^2 = 64\)

5. Add them: \(36 + 64 = 100\)

6. Root the total: \(L = \sqrt{100} = 10\)

The length of the line segment is 10 units.

Common Mistake to Avoid:

Never add the differences before squaring them! You must square the horizontal difference and the vertical difference separately before adding. Squaring first is what turns the two sides of the right-angled triangle into area, as required by Pythagoras.

Section 2: The Midpoint Formula

Finding the midpoint is much simpler than finding the length. The midpoint, \(M\), is the exact centre point of the line segment. To find the centre, we simply calculate the average of the x-coordinates and the average of the y-coordinates.

2.1 The Concept: Finding the Average Position

If your lowest score on a test was 50% and your highest was 90%, the average score is \(\frac{50+90}{2} = 70\%\). The midpoint works exactly the same way: it's the average location.

2.2 The Official Midpoint Formula

The midpoint (\(M\)) of the line segment between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the coordinate pair:

$$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$$

Notice the output is a coordinate \((x, y)\), not just a single number!

Step-by-Step Guide for Midpoint Calculation

1. Add the x-coordinates: Find the sum: \((x_1 + x_2)\).
2. Divide the x sum by 2: This gives you the x-coordinate of the midpoint.
3. Add the y-coordinates: Find the sum: \((y_1 + y_2)\).
4. Divide the y sum by 2: This gives you the y-coordinate of the midpoint.
5. Write the Answer: Present the result as a coordinate pair \((x, y)\).

Memory Trick: Midpoint is the Mean (average). Always Add and Divide by 2 for both coordinates.

Example Calculation (Midpoint)

Find the midpoint of the line segment connecting P (-4, 7) and Q (10, -1).

1. Add x's: \(-4 + 10 = 6\)

2. Divide x sum by 2: \(\frac{6}{2} = 3\)

3. Add y's: \(7 + (-1) = 6\)

4. Divide y sum by 2: \(\frac{6}{2} = 3\)

5. The midpoint M is (3, 3).

Did you know? You can use the midpoint formula to check if a point lies exactly halfway between two others! If a point is claimed to be the centre, check if its coordinates match the formula result.

Common Mistake to Avoid:

When calculating midpoint, you Add the coordinates. Students sometimes accidentally subtract them (like in the distance formula) or forget to divide by 2. Midpoint is about finding the middle point, so addition and division by two are essential.