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### Course: High school geometry>Unit 4

Lesson 7: Solving modeling problems with similar & congruent triangles

# Geometry word problem: the golden ratio

Triangle similarity, the Golden ratio, and art, all converge in this inspiring video! Created by Sal Khan.

## Want to join the conversation?

• I know this looks complicated, but:
Since φ +1= φ^2,
that would mean that (φ +1)^2= (φ ^ 2)^2=φ ^ 4 right?

Also, (φ +1)^2= (φ +1)(φ +1)=φ ^2+2φ +1
So, φ^4= φ ^2+2φ +1

And since φ^2=φ+1,
Then φ^4= (φ+1)+2φ+1=3φ+2

Did I make a mistake in there somewhere?

Also, does this mean that φ raised to any power can be simplified to a linear equation in the form ax+b?
• Your math is all correct. But, I don't think you can simplify just any power of φ as you suggest.

Also, remember that φ is constant, not a variable, so whatever power you put it to (other than a variable) would result in a constant, not a linear equation.
• How do we know for a fact that the upper endpoint of the red middle line should be at the arc and not on top of its head? If the point is at his head, the golden ratio wouldnt line up with his eyes. Did rembrandt use the golden ratio or did we fit it in?
• My thoughts also. Perhaps Sal has seen some of Rembrandt's notes that would tell us that he intentionally used the golden ratio. But perhaps he hasn't seen those notes or there were no notes. Maybe Rembrandt did it this way because it looked right to him. I think it is dangerous to claim to know what an artist probably did without evidence.
• Is the golden ratio really found in all these old paintings?
• The occurrence of the golden ratio is greatly overstated in nature as well as in art and architecture. If something is famous, it is not that hard to play around with finding a ratio near 1.6 and proclaim the golden ratio for almost anything. And those who are fond of seeing the golden ratio everywhere can be very lax about how close to about 1.618 something has to be (the golden ratio has even been proclaimed when the actual ratio was closer to 1.4).

There are artworks that do make use of the golden rectangle / golden ratio, but it not really all that commonplace. I have not studied the claims about Rembrandt and the golden ratio, but it is certain that Salvador Dali did use it. The claims about Leonardo da Vinci using it are not well supported, and especially not in the Mona Lisa.
• At , why are corrseponding altitudes at similar triangles have the same ratio as other corresponding parts such as corresponding sides? perhaps i missed the part where we proves or speaks about corresponding altitudes having the same ratio of the sides? how can we prove this?
• This was proven in Sal's videos on similarity.
• The entire base ratio, CD:BC, is equal to phi:1. The entire altitude ratio, AC:AQ, is equal to phi + 1:1. If you wanted to find the ratio of, AB:AP, in terms of phi, how would you know which value or ratio of phi to use. Would it be phi:1 or phi+1:1. What determines which to use? Thank You.
• I think because AC:AQ refers to the two different triangles, that is the ratio we use as we know the triangles are similar already. whereas CD:BC is just the ratio of the base parts of the larger triangle...
• what is the golden ratio