A geometry problem by Jilvan Júnior

Geometry Level 3

If the point O is the origin of the axis E and A is the point in this axis where have coordinate 1. Who is the coordinate of the point X which divide the segment OA in Golden ratio?

1.6180339887... Poteto 2 1 + 5 \frac {2} {1 + \sqrt{5}} 1 + ( 5 ) 2 \frac {1 + \sqrt(5)} {2}

Jilvan Júnior by Jilvan Júnior , 26, Brazil

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1 solution

Let the coordinate of X be x . Then according to question, \text{Let the coordinate of } X \text{ be } x. \text{ Then according to question, }

O X X A = φ \large\frac{OX}{XA} = \varphi

x 1 − x = φ \large\frac{x}{1-x} = \varphi

x = ( 1 − x ) φ ⇒ ( 1 + φ ) x = φ ⇒ x = φ 1 + φ = 1 + 5 3 + 5 = ( 1 + 5 ) ( 3 − 5 ) ( 5 + 1 ) ( 3 + 5 ) ( 3 − 5 ) ( 5 + 1 ) = 2 1 + 5 \large x = (1-x)\varphi \newline\large\Rightarrow (1+\varphi)x = \varphi\newline\large\Rightarrow x = \frac{\varphi}{1+\varphi} = \frac{1+\sqrt{5}}{3+\sqrt{5}} = \frac{(1+\sqrt{5})(3-\sqrt{5})(\sqrt{5} + 1)}{(3+\sqrt{5})(3-\sqrt{5})(\sqrt{5} + 1)} = \frac{2}{1+\sqrt{5}}

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