Coordinate Geometry

The distance formula: Pythagoras on a grid

To find the distance between (x₁, y₁) and (x₂, y₂), draw a right triangle under the segment. Its legs are the horizontal change Δx = x₂ − x₁ and the vertical change Δy = y₂ − y₁. By the Pythagorean theorem:

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Example. From (−2, 1) to (4, −2): Δx = 6, Δy = −3, so d = √(36 + 9) = √45 = √(9 · 5) = 3√5 ≈ 6.71. The square makes a negative change positive, so the order of the points does not matter.

Exact answers. Leave a square root unless it comes out whole, and simplify it by taking out square factors: √72 = √(36 · 2) = 6√2.

The midpoint formula

The midpoint of a segment averages the coordinates of its endpoints:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Working backwards. If M is the midpoint and you know one endpoint A, the other endpoint is B = (2x_M − x_A, 2y_M − y_A). Whatever step takes you from A to M takes you from M to B.

Slope, parallel and perpendicular

The slope of the line through two points is m = (y₂ − y₁)/(x₂ − x₁), rise over run. A horizontal line has slope 0; a vertical line has an undefined slope, because its run is 0. (Graphing lines from their equations is covered in Algebra: Graphing Linear Equations.)

• Parallel lines have equal slopes (or are both vertical).
• Perpendicular lines have slopes whose product is −1, so each is the negative reciprocal of the other: 2/3 and −3/2. A horizontal line is perpendicular to a vertical one.

Why −1? Turn the slope triangle with run 3 and rise 2 through 90°. The run and rise swap places and one changes sign: run −2, rise 3, so the new slope is 3/(−2) = −3/2.

The equation of a circle

A circle is every point at distance r from its center. Put the center at (h, k); a point (x, y) is on the circle exactly when the distance formula gives r. Squaring both sides:

(x − h)² + (y − k)² = r²

• The center of (x + 2)² + (y − 5)² = 49 is (−2, 5), because x + 2 = x − (−2). The radius is √49 = 7, not 49.
• A circle centered at the origin is x² + y² = r².
• To test a point, substitute it: (7, 7) is on the circle in the figure because (7 − 4)² + (7 − 3)² = 9 + 16 = 25.

Coordinate proofs

In a coordinate proof you show a fact about a figure by computing with its coordinates. The three tools do different jobs:

• Slope tests parallel (equal slopes) and perpendicular (product −1).
• Distance tests equal lengths.
• Midpoint tests whether segments bisect each other: a quadrilateral is a parallelogram exactly when its diagonals share a midpoint.

Classifying a quadrilateral. For A(1, 1), B(5, 2), C(4, 6), D(0, 5): slopes AB = DC = 1/4 and BC = AD = −4, so both pairs of opposite sides are parallel (a parallelogram), and (1/4)(−4) = −1, so the corners are right angles (a rectangle). All four sides are √(4² + 1²) = √17, so ABCD is a square. Name the most specific shape the evidence proves.

Showing a triangle is right. For P(0, 0), Q(4, 2), R(3, 4): slope PQ = 1/2 and slope QR = 2/(−1) = −2, and (1/2)(−2) = −1, so the right angle is at Q. As a check with the converse of the Pythagorean theorem: PQ² + QR² = 20 + 5 = 25 = PR².

Choosing coordinates. To prove something about every figure of a kind, place it conveniently and use letters: a rectangle at (0, 0), (a, 0), (a, b), (0, b). Its diagonals both have midpoint (a/2, b/2), so the diagonals of every rectangle bisect each other.

Common mistakes

• Mixing up the order in slope. Subtract in the same order top and bottom: (y₂ − y₁)/(x₂ − x₁), never (y₂ − y₁)/(x₁ − x₂).
• Run over rise. Slope is change in y over change in x.
• Sign errors in a circle’s center. (x − 3)² + (y + 1)² = 16 has center (3, −1).
• Forgetting the square root. In a circle’s equation the right side is r², and in the distance formula √(Δx² + Δy²) is not Δx + Δy.
• Perpendicular slopes. The negative reciprocal of 3 is −1/3, not −3 and not 1/3.
• Stopping too early. A rectangle with four equal sides is a square; give the most specific name.

History: Descartes, Fermat and the marriage of algebra and geometry

In 1637 René Descartes published his Discours de la méthode at Leiden with three appendices, one of them La Géométrie. In it he turned geometric problems into equations about lengths and solved them with algebra. That is why coordinates are still called Cartesian, after the Latin form of his name, Cartesius. His axes were not quite ours: the habit of drawing two perpendicular number lines with negative values came later.

At about the same time, Pierre de Fermat, a lawyer in Toulouse, reached the same idea on his own. Working from Apollonius’ lost book on Plane Loci, he wrote Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which showed that an equation in two unknowns describes a line or curve. He sent it to mathematicians in Paris, but never polished it for print, so Descartes got the public credit.

Sources

• René Descartes, La Géométrie (Leiden, 1637), French text at Project Gutenberg.
• MacTutor History of Mathematics, University of St Andrews: René Descartes (biography, with the 1637 Discours and its appendix La Géométrie); Pierre de Fermat (biography, with his restoration of Apollonius’ Plane Loci and his Introduction to Plane and Solid Loci).

Credits: all diagrams on this page are original SVG drawings made for Jugami.