Golden ratio

There are 16 NRICH Mathematical resources connected to Golden ratio
Golden Thoughts
problem
Favourite

Golden thoughts

Age
14 to 16
Challenge level
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Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.
Pent
problem

Pent

Age
14 to 18
Challenge level
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The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Golden Ratio
problem

Golden ratio

Age
16 to 18
Challenge level
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Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.
Pentakite
problem

Pentakite

Age
14 to 18
Challenge level
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Given a regular pentagon, can you find the distance between two non-adjacent vertices?
Golden Fibs
problem

Golden fibs

Age
16 to 18
Challenge level
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When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!
Golden Fractions
problem

Golden fractions

Age
16 to 18
Challenge level
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Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
Gold Yet Again
problem

Gold yet again

Age
16 to 18
Challenge level
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Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."
Gold Again
problem

Gold again

Age
16 to 18
Challenge level
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Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.
Pythagorean Golden Means
problem

Pythagorean golden means

Age
16 to 18
Challenge level
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Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.
Golden Triangle
problem

Golden triangle

Age
16 to 18
Challenge level
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Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.