The Golden Ratio has been a recurring star on this blog ever since we first explored its mysterious beauty years ago. If you missed that journey, be sure to click here for the original article.
Its magic did not end there. We went on to solve a whole collection of fascinating problems based on the Golden Ratio, each revealing yet another surprising facet of this extraordinary number. You can revisit those collection by clicking here.
More recently, we explored the elegance of Ptolemy’s Theorem and walked through its proof in detail. Now comes the exciting part!
Today, we shall bring these two mathematical gems together. With a single, elegant application of Ptolemy’s Theorem, we will derive one of the most celebrated results in geometry: the ratio of the length of a diagonal of a regular pentagon to the length of its side. As you will soon discover, that ratio is none other than the magnificent Golden Ratio which is equal to (√5 +1)/2 = 1.618.
Please note that any regular polygon can be inscribed in a circle. So any four vertices of a regular pentagon taken in order will be forming a Cyclic Quadrilateral. And you know the famous Ptolemy's Theorem for cyclic quadrilaterals.
Now, let us draw the diagram by hand. (Computer-generated figures may be perfect, but handwritten sketches possess a soul of their own. Besides, I would much rather leave this world having left not only erasable footprints on the dust of Indoree streets, but fingerprints also on the pages I cherished!) So here is the Diagram and the derivation.
The diagram is self explanatory.
By applying this theorem, one arrives at a relationship of such striking elegance and simplicity that, once encountered, it lingers in the mind forever; you could scarcely forget it, even if you tried.
Though seldom encountered beyond the confines of mathematical circles; this elegant property is a hidden treasure: one that rewards serious puzzle lovers and proves immensely useful for students preparing for competitive examinations such as XAT, CAT, and their counterparts.
Finally, let us apply this knowledge of Golden Ratio to solve this quiz in circulation in quiz groups:
The problem is stated as below: Find value of x without use of Trigonometry/ Trigonometric Tables.
While this problem can be solved by various geometrical methods, we will solve it with the help of above property of regular Pentagon that the ratio of the Diagonal to the side is equal to Golden Ratio which is equal to (√5 +1)/2 = 1.618.
Angle 36° is naturally found in Regular Pentagons. Let the triangle AFE given in the problem be a part of a regular pentagon of side 2 as shown in the attached image.
In the diagram, AE=2, so BE will be equal to 2x (√5 +1)/2 = √5+1. Hence FE or x will be equal to (√5+1)/2.
Thus we can find the value of x instantly after seeing the problem!
Similarly this problem can be solved in the similar manner by constructing 36°-72°-72° or 36°-36°-108° triangle which can be found in Regular Pentagons. (Click here to know more about
36°-72°-72° triangle. And click here to know more about
36°-36°-108° triangle.).
