It is a enjoyable downside I noticed on Reddit AskMath.
What’s the reply?
As standard, watch the video for an answer.
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Reply To Ratio sq. aspect to chord
(Just about all posts are transcribed shortly after I make the movies for them–please let me know if there are any typos/errors and I’ll appropriate them, thanks).
Assemble the diagonal of the sq. which is able to move by way of the middle of the circle. The 2 sq. sides and hypotenuse type a 45-45-90 proper triangle, so the diagonal is √2 the size of a aspect. Thus the diagonal is a√2, and half of that’s (a√2)/2.
The triangle with one acute angle of 75 levels should have one other acute angle of 15 levels. And the angle between that and the diagonal will likely be 45 – 15 = 30 levels.
Assemble the perpendicular bisector to the chord with size b. That is the shorter leg of a 30-60-90 proper triangle whose hypotenuse is half the diagonal (a√2)/2. The shorter leg will likely be half of that, or (a√2)/4.
Lastly assemble a radius to the endpoint of the chord with size b. The radius of the circle is half the sq. aspect, so it’s a/2.
We now have a proper triangle with legs (a√2)/4, b/2, and a hypotenuse a/2. So we now have:
[(a√2)/4]2 + (b/2)2 = (a/2)2
2a2/16 + b2/4 = a2/4
2a2 + 4b2 = 4a2
4b2 = 2a2
2 = a2/b2
√2 = a/b
(Within the final step we solely took the constructive sq. root as we’re coping with distances)
Thus a/b = √2.
Reference
https://www.reddit.com/r/askmath/feedback/1kx9gkr/hey_guys_can_you_help_me_with_geometry/
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