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Small Geometry Puzzle
Challenge: write up your initial instinct, then try to prove it and let me know what happens. :D
I really like this puzzle. And I'm sure that there's a simpler proof than the one I'm using right now.
Puzzle:
Solution:
The orange is exactly the same area as the sum of the two blue areas.
Proof:
Step 1: The Orange circle is exactly 1/4 of the big circle. Proof by picture:
(Draw another equlateral triangle inscribed in the circle, with a 180 degree rotation relative to the original triangle. These two triangles overlap in a regular hexagon that can be split into 6 triangles like a pie. Each of these triangles is the same size as any of the 6 points of the star now inscribed in the circle. So it is clear that the radius of the orange circle is half the radius of the large circle. Since A = pi*r^2, this means that the area of the orange circle is 1/4 the area of the whole circle.
Step 2: Blue area is exactly 1/3 of the remaining area in the large circle that is not part of the orange circle. Proof by picture:
The remaining area has three semetrical components.
Step 3: Since the orange circle is 1/4 of the large circle, the remaining interior of the large circle is 3/4 of the area. The blue area is 1/3 of that, and 1/3 of 3/4 is 1/4. So both the orange region and the combined two blue regions are 1/4 the area of the whole circle.
Boom! Done!
Using this to draw a very pretty infinite summation picture. :)
1/4 + 1/16 + 1/64 + ... 1/4^n as n approaches infinity = 1/3
I really like this puzzle. And I'm sure that there's a simpler proof than the one I'm using right now.
Puzzle:
Solution:
The orange is exactly the same area as the sum of the two blue areas.
Proof:
Step 1: The Orange circle is exactly 1/4 of the big circle. Proof by picture:
Step 2: Blue area is exactly 1/3 of the remaining area in the large circle that is not part of the orange circle. Proof by picture:
Step 3: Since the orange circle is 1/4 of the large circle, the remaining interior of the large circle is 3/4 of the area. The blue area is 1/3 of that, and 1/3 of 3/4 is 1/4. So both the orange region and the combined two blue regions are 1/4 the area of the whole circle.
Boom! Done!
Using this to draw a very pretty infinite summation picture. :)
1/4 + 1/16 + 1/64 + ... 1/4^n as n approaches infinity = 1/3
no subject
Components of problem solving - like picking which length to set as the 'unit' of measurement are relevant not only in math, right? Hmm... in fact, it strikes me as something that could even be relevant to analysis approaches in the humanities. :)
And in a lot of geometry problems, your first strategy would have worked out ---
"A is equal to B, which is equal to C, which is greater than D, so therefore A is greater than D as well..." or something like that.
I like turning this proof around into a construction too. Aka, now you know one way to draw a smaller circle inside a larger circle so that they have the same center and the small circle is exactly 1/4 the area of the large circle. Although... maybe there's an easier way than inscribing a triangle inside the circle and then a circle inside the triangle. ;-P
no subject
Well, along the way we proved that the smaller circle's radius is half the size of the larger one's. So to construct it without the triangle, you'd just make a circle, draw a radius, and then draw the smaller circle so its perimeter intersected that first radius exactly half way.
Components of problem solving - like picking which length to set as the 'unit' of measurement are relevant not only in math, right?
Oh yeah. "What do I measure and how?" is perhaps the biggest practical recurring problem in my life.