# Resources tagged with: Golden ratio

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Broad Topics > Fractions, Decimals, Percentages, Ratio and Proportion > Golden ratio ### Golden Triangle

##### Age 16 to 18 Challenge Level:

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. ##### Age 16 to 18

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio? ### Pentakite

##### Age 14 to 18 Challenge Level:

ABCDE is a regular pentagon of side length one unit. BC produced meets ED produced at F. Show that triangle CDF is congruent to triangle EDB. Find the length of BE. ### Golden Ratio

##### Age 16 to 18 Challenge Level:

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi. ### Leonardo of Pisa and the Golden Rectangle

##### Age 7 to 16

Leonardo who?! Well, Leonardo is better known as Fibonacci and this article will tell you some of fascinating things about his famous sequence. ### Pythagorean Golden Means

##### Age 16 to 18 Challenge Level:

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. ### Darts and Kites

##### Age 14 to 16 Challenge Level:

Explore the geometry of these dart and kite shapes! ### Golden Powers

##### Age 16 to 18 Challenge Level:

You add 1 to the golden ratio to get its square. How do you find higher powers? ### Pent

##### Age 14 to 18 Challenge Level:

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 Fractions

##### Age 16 to 18 Challenge Level:

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers. ### Golden Fibs

##### Age 16 to 18 Challenge Level:

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio! ### Gold Again

##### Age 16 to 18 Challenge Level:

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72. ### Golden Thoughts

##### Age 14 to 16 Challenge Level:

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. ### Whirling Fibonacci Squares

##### Age 11 to 16

Draw whirling squares and see how Fibonacci sequences and golden rectangles are connected. ### Gold Yet Again

##### Age 16 to 18 Challenge Level:

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection." ### The Golden Ratio, Fibonacci Numbers and Continued Fractions.

##### Age 14 to 16

An iterative method for finding the value of the Golden Ratio with explanations of how this involves the ratios of Fibonacci numbers and continued fractions.