Cartesian Plane & Distance Formula

The Cartesian plane is like a map with two number lines crossing at right angles. Just as GPS coordinates help you find any location in Nepal, the Cartesian coordinate system lets you pinpoint any location in math.

The Cartesian Plane

The horizontal line is the x-axis and the vertical line is the y-axis. They meet at the origin (0, 0). Every point is described by an ordered pair (x, y).

The plane is divided into four quadrants:

• Quadrant I: x > 0, y > 0 (top-right)
• Quadrant II: x < 0, y > 0 (top-left)
• Quadrant III: x < 0, y < 0 (bottom-left)
• Quadrant IV: x > 0, y < 0 (bottom-right)

Distance Between Two Points

Distance Formula: For two points A(x1, y1) and B(x2, y2), d = root of [(x2 - x1)^2 + (y2 - y1)^2]

This comes directly from the Pythagorean theorem.

Worked Example: Find the distance between A(1, 2) and B(4, 6).

d = root of [(4-1)^2 + (6-2)^2] = root of [9 + 16] = root of 25 = 5 units

Midpoint Formula

The midpoint is the point exactly halfway between two endpoints.

Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)

Worked Example: Find the midpoint of P(2, 3) and Q(8, 7).

M = ((2+8)/2, (3+7)/2) = (5, 5)

Section Formula

If a point divides the line joining A(x1, y1) and B(x2, y2) in the ratio m:n, then:

Section Formula: P = ((m.x2 + n.x1)/(m+n), (m.y2 + n.y1)/(m+n))

Worked Example: Find the point that divides the segment joining A(1, 2) and B(7, 8) in the ratio 2:1.

P = ((2 x 7 + 1 x 1)/3, (2 x 8 + 1 x 2)/3) = (15/3, 18/3) = (5, 6)

Did You Know? The Cartesian plane is named after French mathematician Rene Descartes. Coordinate geometry bridges algebra and geometry -- you can solve geometric problems using algebraic equations.

Key Takeaways

• Every point on the Cartesian plane has coordinates (x, y)
• The distance formula is based on the Pythagorean theorem
• The midpoint formula averages the coordinates
• The section formula generalizes the midpoint for any ratio