Co-ordinate Geometry

282. The Four Quadrants · mapping signs to the coordinate plane

The signs of P's coordinates always match its quadrant's convention.

The axes divide the plane into four quadrants. Reading anticlockwise from the top right, their sign patterns are I (+, +), II (−, +), III (−, −), and IV (+, −) — so a point's signs reveal its quadrant.

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Selina ICSE: Co-ordinate Geometry

What this lesson covers

Try to break it

Drag P into each region in turn. Quadrant I: (+, +). Quadrant II: (−, +). Quadrant III: (−, −). Quadrant IV: (+, −). On the axes one coordinate is 0; at the origin both are 0. Try to find a point with both coordinates positive outside Quadrant I; impossible.

How you build it

Plot one point in each of the four quadrants.

Point tool: place A in Quadrant I (top-right) — both x and y positive (+, +).
Point tool: place B in Quadrant II (top-left) — x negative, y positive (−, +).
Point tool: place C in Quadrant III (bottom-left) — both x and y negative (−, −).
Point tool: place D in Quadrant IV (bottom-right) — x positive, y negative (+, −). All four quadrants done!

The proof, step by step

Prove that the signs of the coordinates of a point determine its quadrant.

The coordinate axes divide the plane into four quadrants.
In Quadrant I, both x and y are positive.
In Quadrant II, x is negative and y is positive.
In Quadrant III, both x and y are negative.
In Quadrant IV, x is positive and y is negative.

Worked example

A point P has coordinates (−5, 3). In which quadrant does P lie?

Since the x-coordinate is negative and the y-coordinate is positive, the point lies in the second quadrant (X'OY).

First Quadrant
Second Quadrant — correct
Third Quadrant
Fourth Quadrant