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" [math]\varphi[/math], that is one of the fundamental math constants.[br][br]Italian version available [url=https://www.geogebra.org/m/ScyvKdfm#material/twj8tuqt]here[/url].
Practice Zone
When the construction is finished, use the [i]Distance[/i] tool of GeoGebra and measure the segments [i]AB,[/i] [i]AC [/i]and [i]CB.[/i][br]Move points [i]A [/i]and[i] B [/i]and write down the measures of the segments, then create a table containing a sample of measures of segment [i]AB[/i], the corresponding measures of segments[i] AC [/i]and[i] CB[/i], and the ratios [math]\frac{AB}{AC}[/math] and [math]\frac{AC}{CB}[/math].[br][br]What do you observe?
From Geometry to Algebra
Let's "translate" into algebraic terms the geometric proportion [math]AB:AC=AC:CB[/math] that defines the golden ratio.[br]Let [math]a[/math] be the length of the segment [math]AB[/math], and [math]x[/math] the length of the golden section [math]AC[/math].[br]Then, [math]CB=AB-AC=a-x[/math].[br][br]Rewriting the proportion using the algebraic notation, we obtain [math]a:x=x:\left(a-x\right)[/math].[br][br]The product of means equals the product of the extremes, therefore [math]x^2=a\left(a-x\right)[/math].[br]This is a quadratic equation: expanding and reducing to normal form we get [math]x^2+ax-a^2=0[/math] that we can solve using the quadratic formula, obtaining [math]x_{1,2}=\frac{-a\pm\sqrt{a^2+4a^2}}{2}=\frac{-a\pm a\sqrt{5}}{2}[/math].[br][br]Since this solution refers to the length of a segment, it must be positive. Discarding the negative solution and collecting the common term [math]a[/math] we have [math]x=\frac{a\left(\sqrt{5}-1\right)}{2}[/math].[br]We have calculated the length of the golden section [math]x=AC[/math] of a segment [math]AB[/math] that is [math]a[/math] units long.[br][br]We can now calculate the golden ratio [math]\frac{AB}{AC}=\frac{a}{\frac{a\left(\sqrt{5}-1\right)}{2}}[/math].[list][*]Reducing the fraction by multiplying the numerator by the reciprocal of the denominator, and simplifying the result we have [math]=a\cdot\frac{2}{a\left(\sqrt{5}-1\right)}=\frac{2}{\sqrt{5}-1}[/math][/*][*]Rationalizing and simplifying the result we obtain: [math]=\frac{2}{\sqrt{5}-1}\cdot\frac{\sqrt{5}+1}{\sqrt{5}+1}=\frac{2\left(\sqrt{5}+1\right)}{5-1}=\frac{2\left(\sqrt{5}+1\right)}{4}=\frac{\sqrt{5}+1}{2}[/math].[/*][/list][br]Therefore the golden ratio is [math]\varphi=\frac{\sqrt{5}+1}{2}[/math].[br]