Geometry: Similar Triangles

MA.912.D.11.5 Explore and use other sequences found in nature such as the

Fibonacci sequence and the golden ratio.

Block 30

Fibonacci sequence

• In mathematics, the Fibonacci numbers are the numbers in the following sequence:

1, 1, 2, 3, 5, 8, 13, 21 …• By definition, the first two Fibonacci numbers

are 1 and 1, and each remaining number is the sum of the previous two.

Fibonacci sequence: recurrence

• In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

• with seed values

12 nnn FFF

1,1 21 FF

Graph of consecutive values of Fibonacci sequence

Golden mean

• Ratios of consecutive values of Fibonacci sequence:

• Fn+1/Fn

• Tend to a number – it is Golden Mean

Ratio of terms of Fibonacci sequence converge to the Golden mean

Approximation of Golden Rectangle

Golden triangle

Golden triangle

A golden triangle is an isosceles triangle in which the two longer sides have equal lengths and in which the ratio of this length to that of the third, smaller side is the golden ratio:

2

51

b

a

Golden triangle

• Different description of a golden triangle is an isosceles triangle with angles 360, 720 and 720

Golden triangle: examples

• As an example of the appearance of Golden Triangles: the outside triangles of a pentagram are Golden Triangles.

Golden triangle: properties

• Position of Golden Triangles in regular pentagon

Golden rectangle, golden triangle and logarithmic spirals

Logarithmic and golden spiral

• In the logarithmic spiral the distances between the turnings increase in geometric progression

• In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio

Approximation to Golden spiral

Steps in construction of logarithmic spiral

• ABC golden triangle• ABD similar triangle to

ABC – golden triangle

Steps in construction of aproximation of logarithmic spiral • ABC golden triangle• ABD similar triangle to

ABC – golden triangle• We will continue to

construct similar triangles in this fashion

Can you recognize golden triangles?

Approximation of logarithmic spiral

Class discussion:

• Ask students to search the internet for Golden Ratio in Art and Architecture

• Groups presentation and discussion: how to enhance teaching with connection to art and architecture

• Holy Family by Michelangelo: