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## Geometry (all content)

### Course: Geometry (all content)>Unit 12

Lesson 6: Solving modeling problems with similar and congruent triangles

# Geometry word problem: Earth & Moon radii

Similarity and the Golden ratio join forces again, this time to save the Earth! (Or maybe just finding its radius). Created by Sal Khan.

## Want to join the conversation?

• hello, why is it 6371 km square root of phi, but not square root of phi minus 6371 km. thanks.
• Yeah, that's tricky. I would look at it in three steps (where Sal combined it into one step). (1)The triangle we are given (the one drawn in the picture) is in units of earth-radii. Segment EL, for instance, is 1 earth radius - equal to the radius of earth. Segment ME, conveniently the length of both an earth and a moon radius, is given as sqrt(phi) earth radii (or aproximately 1.27... earth-radii).

(2) So then when you draw your triangle in kilometers, the base is 6371 km, and the height is 6371 km + r km (where r is the radius of the moon). (3) Since you have two congruent triangles, you can create a ratio: 6371/1 = (6371+r)/sqrt(phi). Cross multiply and you get: 6371(sqrt(phi)) = (6371 + r)(1) -- which catches you up to Sal.
• The problem says that the golden ratio is the only number that has the property of its square being one more than itself. How did they prove this? or is it even proven?
• It can be proven simply by solving for an unknown value x that satisfies the conditions. In other words, solving for x in x^2 = x + 1. This ends up just being a quadratic (x^2 - x - 1 = 0) that you can use the quadratic formula on. The only numbers that satisfy the equation are the golden ratio and one minus the golden ratio (1.618.... and -0.618....)
• How did he get 6371 times the square root of phi?
• The base was 1 as you could see on the picture and then it was proportioned so you got the actual length which is 6371 km as Sal drew it. This is the same as saying 6371 * 1. So logically the same should follow for the other side which is sqrt(phi) so you got 6371 * sqrt(phi).
• I'm failing to understand even the terms of this problem. How is the side the is the square root of phi also the square root of phi*one earth radii? I just can't see why Sal multiplied 6371km by the square root of phi in his new triangle. Please help!
• Let me try to help here.

The units were in earth radii which was one unit of Earth Radii, so Sal changed them to km by multiplying them by Earth's radii in terms of kilometers. He did this to all the numbers, in the equation of a^2+b^2=c^2, otherwise the equation would have been changed.

I hope I helped, kind of new to this myself.
• Phi squared is `phi + 1`... Is the square root of phi anything special?
• No matter how many times you take the square root of 1, you will always get 1. Similarly, if you recursively take the square root of a value x, it will approach 1. If x is positive it will diminish to 1, if 0 < x < 1 then x will increment up to the limit of 1. Not exclusive to phi, but a cool property of recursive square roots for this number range.
• If the radius of the moon and the radius of the earth are in proportion to φ, is the moon (in any way) and any other satellite (artificial or manmade) proportion to φ? Thanks in advance.
• It's possible this Moon:Earth ratio was luck. However, our best theory is that the Moon was formed via a planet-planet collision. So, it's also possible that when two planets collide, and are similar enough in size to create a moon at all, that it's biased to be in a ratio like ours is.
• How can Phi go on forever and ever?
• Is it a non-repeating non terminating number. The definition of this is that the decimals in the number go on for an infinite amount. Other example of this is π and √2.
• Hi, at please tell me how did he get (sqrt phi-1)?
• He factored out a 6371:
(6371√φ) - 6371 = 6371([√φ] - 1)
Note that when you distribute the 6371 on the right side of the equation, you get back to the left side of the equation. Therefore, both are equal.
• Where can I find the video on golden triangle
• This video is about the golden triangle :). If you want to learn about the golden ratio the previous video will do.
(1 vote)
• I'm confused about how he got the first measurements. How does a^2 = the square root of phi?
• He set up the pythagorean theorem as the analogous equation.
a^2 + b^2 = c^2
Thus since phi + 1 = phi^2, one can conclude that
a^2 = phi <=> √(a^2) = √phi <=> a = √phi