Golden Ratio and Dividing a Segment in a Given Ratio

Dividing a segment in a given ratio m : n means finding the point that splits it into two parts with lengths in that ratio, and the golden ratio φ = (1 + √5)/2 ≈ 1.618 is the special ratio in which the whole is to the larger part as the larger part is to the smaller. In brief: a segment of length 10 divided in the ratio 2 : 3 has parts 4 and 6. A segment of length 10 divided in the golden ratio has parts of about 6.18 and 3.82.

Dividing a segment in a ratio is a basic skill in geometry and in coordinate geometry, used for locating points on a line. The golden ratio is a special case with unusual algebraic and geometric properties, and it appears in pentagons and in the Fibonacci numbers. This guide explains how to divide a segment in a given ratio by calculation, by the section formula and by construction, and how to find and construct the golden ratio.

How do you divide a segment in a given ratio by calculation?

To divide a segment of length L in the ratio m : n, split it into m + n equal parts. The first part has length mL/(m + n) and the second has length nL/(m + n).

Example 1

Divide 35 cm in the ratio 2 : 5. The total number of parts is 7, and one part is 5 cm. The pieces are 10 cm and 25 cm. Check: 10 : 25 = 2 : 5 ✓ and 10 + 25 = 35 ✓.

Example 2: a three-way split

Divide 60 m in the ratio 1 : 2 : 3. There are 6 parts of 10 m each, so the pieces are 10 m, 20 m and 30 m.

What is the section formula in coordinates?

The point P that divides the segment from A(x₁, y₁) to B(x₂, y₂) internally in the ratio m : n (that is AP : PB = m : n) has the coordinates

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

For m = n = 1 this is the midpoint formula.

Example 3

A(1, 2) and B(11, 7), ratio AP : PB = 3 : 2. Then P = ((2 · 1 + 3 · 11)/5, (2 · 2 + 3 · 7)/5) = (35/5, 25/5) = (7, 5). Check: AP = √(36 + 9) = √45 and PB = √(16 + 4) = √20. The ratio is √45 : √20 = 3√5 : 2√5 = 3 : 2 ✓.

Example 4: external division

For an external division in the ratio m : n, P lies on the line outside the segment, and the formula is ((−n x₁ + m x₂)/(m − n), (−n y₁ + m y₂)/(m − n)). For A(0, 0), B(4, 0) and ratio 3 : 1: P = (3 · 4/2, 0) = (6, 0), and AP = 6, PB = 2, which has ratio 3 : 1 ✓.

How do you divide a segment in a given ratio by construction?

1. Draw the segment AB. From A draw a ray at any angle, not along AB.
2. On the ray, mark m + n equal steps with the compass: A₁, A₂, …, A_m+n.
3. Join A_m+n with B.
4. Through A_m draw a line parallel to A_m+nB. It meets AB at the point P that divides AB in the ratio m : n.

The method works by the intercept theorem: parallel lines cut proportional segments on the two lines. The parallel line is drawn by copying an angle.

What is the golden ratio?

A segment is divided in the golden ratio if the whole segment is to the larger part as the larger part is to the smaller part. If the larger part is a and the smaller is b, then (a + b)/a = a/b. Let φ = a/b. Then 1 + 1/φ = φ, so φ² − φ − 1 = 0, and the positive solution is

φ = (1 + √5)/2 ≈ 1.6180339887…

Properties: φ² = φ + 1 ≈ 2.618, 1/φ = φ − 1 ≈ 0.618. For a segment of length L, the larger part is L/φ ≈ 0.618L and the smaller part is L(1 − 1/φ) = L/φ² ≈ 0.382L. For L = 10 the parts are 6.18 and 3.82. The number φ is irrational.

The ratio of consecutive Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … approaches φ: 55/34 ≈ 1.6176. In a regular pentagon, the ratio of a diagonal to a side is exactly φ. A golden rectangle has sides in the ratio φ : 1. If a square is cut from it, the remaining rectangle is again golden.

How do you construct the golden section?

1. Draw the segment AB. Construct the perpendicular at B and mark point C on it with BC = AB/2.
2. Draw the line AC. With center C and radius CB, draw an arc that meets AC at D.
3. With center A and radius AD, draw an arc that meets AB at P.
4. The point P divides AB in the golden ratio: AP : PB = φ : 1.

Check with AB = 2: BC = 1, AC = √5, AD = √5 − 1 ≈ 1.236 = AP. Then PB = 2 − 1.236 = 0.764, and AP/PB = 1.236/0.764 ≈ 1.618 ✓.

Values and properties of the golden ratio

Where do these ideas come from?

Euclid described the division of a segment in “extreme and mean ratio” in the Elements, which is the golden ratio, and used it to construct the regular pentagon. The name “golden ratio” became common in the nineteenth century. Fibonacci’s numbers appeared in his 1202 book Liber Abaci in a problem about rabbits, and their link to φ was noted later, by Johannes Kepler in the seventeenth century. Many claims that the golden ratio appears in famous paintings and buildings are not supported by documents and are disputed by historians.

Where are they used today?

The section formula is used in computer graphics to compute points along a line, in the form of linear interpolation: P = (1 − t)A + tB. The golden ratio is used in some design layouts and in the golden-section search, an algorithm for finding the minimum of a function. Dividing a segment in a ratio is also used by surveyors and in mixing problems.

Which mistakes are common?

Mixing up the order in the section formula

For AP : PB = m : n the weight m goes with B (the far end), not with A. For m : n = 3 : 2, P is closer to B, as in Example 3.

Dividing by m or n instead of m + n

The total number of parts is m + n.

Confusing φ with 1/φ

φ ≈ 1.618 is the ratio of the larger to the smaller part, and 1/φ ≈ 0.618 is the ratio of the larger part to the whole.

Treating 1.6 as exact

The golden ratio is irrational. Approximations such as 1.618 or 8/5 are only approximate.