Orienting Yourself: The Use of Coordinates

Every point in a plane is fixed by two numbers (x, y). From the coordinates we can find quadrants, distances, midpoints and reflections.

Key ideas

The x-axis and y-axis meet at right angles at the origin O (0, 0). The x-coordinate is always written first.
Quadrants go anticlockwise from the top right: I (+, +), II (−, +), III (−, −), IV (+, −).
A point with y = 0 lies on the x-axis; a point with x = 0 lies on the y-axis.
Points with the same y-coordinate lie on a line parallel to the x-axis; with the same x-coordinate, parallel to the y-axis.
Reflect in the x-axis: the y-coordinate changes sign. In the y-axis: the x-coordinate changes sign. In the origin: both change sign.
Use the distance formula to test whether three points form an isosceles, right-angled or equilateral triangle.

Distance between two points	d = √((x₂ − x₁)² + (y₂ − y₁)²)
Distance from the origin	d = √(x² + y²)
Midpoint	((x₁ + x₂)/2, (y₁ + y₂)/2)
Same x (vertical distance)	d = difference of the y-coordinates

Say where each point lies: (3, 9), (8, 6), (−6, 4), (7, 6).

Working: I (+, +), II (−, +), III (−, −), IV (+, −); a 0 puts the point on an axis

Answer: quadrant I; quadrant I; quadrant II; quadrant I