Golden Section of a Line Segment - From Geometry to Algebra
Explore the step by step construction of the golden section of a line segment, then use the construction for the activity displayed below the app, and finally discover how to calculate the "golden number" , that is one of the fundamental math constants.
Italian version available here.
Practice Zone
When the construction is finished, use the Distance tool of GeoGebra and measure the segments AB, AC and CB. Move points A and B and write down the measures of the segments, then create a table containing a sample of measures of segment AB, the corresponding measures of segments AC and CB, and the ratios and . What do you observe?
From Geometry to Algebra
Let's "translate" into algebraic terms the geometric proportion that defines the golden ratio.
Let be the length of the segment , and the length of the golden section .
Then, .
Rewriting the proportion using the algebraic notation, we obtain .
The product of means equals the product of the extremes, therefore .
This is a quadratic equation: expanding and reducing to normal form we get that we can solve using the quadratic formula, obtaining .
Since this solution refers to the length of a segment, it must be positive. Discarding the negative solution and collecting the common term we have .
We have calculated the length of the golden section of a segment that is units long.
We can now calculate the golden ratio .
- Reducing the fraction by multiplying the numerator by the reciprocal of the denominator, and simplifying the result we have
- Rationalizing and simplifying the result we obtain: .