Distance Formula

Take two points A(x1, y1) and B(x2, y2) on a coordinate grid. Drop a horizontal line from A and a vertical line from B so they meet at a corner C. Now ABC is a right-angled triangle. The horizontal side has length (x2 - x1) and the vertical side has length (y2 - y1). By Pythagoras' theorem the hypotenuse AB satisfies AB squared = (x2 - x1) squared + (y2 - y1) squared. Taking the square root gives the distance formula. So it is not a new idea at all, just Pythagoras drawn on a grid.

A very common case is the distance of a point P(x, y) from the origin O(0, 0). Here x1 = 0 and y1 = 0, so the formula becomes OP = root( x squared + y squared ). The two legs of the right triangle are simply the x-coordinate and the y-coordinate of the point. This shortcut saves time and appears often in exams.

Once you can measure any side, you can identify shapes from coordinates alone. Find all the side lengths: if two sides are equal the triangle is isosceles, if all three are equal it is equilateral. To test for a right angle, check whether the two shorter sides squared add up to the longest side squared (the converse of Pythagoras). For four points you can test a quadrilateral: equal sides and equal diagonals point to a square, equal sides with unequal diagonals to a rhombus, and so on.

Three points are collinear (lie on one straight line) if the longest distance between them equals the sum of the other two distances. A point that is equidistant from several given points is found by setting the distances equal and solving the equation; this is how the circumcentre of a triangle, equally far from all three vertices, is located. A point on the x-axis has the form (x, 0) and a point on the y-axis has the form (0, y), which simplifies such problems.