Golden ratio

problem
Favourite

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.
problem

Darts and Kites

Age

14 to 16

Challenge level

Explore the geometry of these dart and kite shapes!
problem

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.
problem

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?
problem

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.
problem

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!
problem

Golden Fractions

Age

16 to 18

Challenge level

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.
problem

Gold yet Again

Age

16 to 18

Challenge level

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."
problem

Pentakite

Age

14 to 18

Challenge level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?
problem

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.