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Pure · Coordinate geometry

Chapter 1 · 4

The idea

Length and midpoint of a line segment

The distance between two points as Pythagoras on the horizontal and vertical gaps (left in exact surd form), and the midpoint as the average of the coordinates — plus working backwards from a known midpoint.

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Pure · Coordinate geometry

Length and midpoint of a line segment

The distance between two points as Pythagoras on the horizontal and vertical gaps (left in exact surd form), and the midpoint as the average of the coordinates — plus working backwards from a known midpoint.

Why it works

Pythagoras in disguise

Drop a right-angled triangle between two points A(x1,y1)A(x_1, y_1) and B(x2,y2)B(x_2, y_2): the horizontal leg is the change in xx, the vertical leg is the change in yy, and the segment ABAB is the hypotenuse. Pythagoras then gives the length:

AB=(x2−x1)2+(y2−y1)2AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

The midpoint is an average

The midpoint is the point exactly halfway along, so each of its coordinates is the average of the two endpoints': M=(x1+x22, y1+y22).M = \left( \frac{x_1 + x_2}{2}, \ \frac{y_1 + y_2}{2} \right). That's an average — you add and halve, you don't subtract. (Subtracting and halving gives you half the gap, which is a useful quantity but not the midpoint.)

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