the longest cord: It diameter is the longest cord in the circle?
- Thread starter shahar
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Yes, diameters are chords of a circle, and each chord of a circle that is not a diameter is shorter than the diameter.
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- Nov 24, 2012
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WLOG, I would parametrize a circle of radius 1, centered at \(\displaystyle \left(0,1\right)\) as follows:
\(\displaystyle x(t)=\sin(2t)\)
\(\displaystyle y(t)=-\cos(2t)+1\)
For \(\displaystyle 0\le t<\pi\)
Hence, the chord length \(\displaystyle d\) will have a length whose square is:
\(\displaystyle d^2=x^2(t)+y^2(t)= \sin^2(2t)+\cos^2(2t)-2\cos(2t)+1= 2(1-\cos(2t))\)
We know:
\(\displaystyle -1\le\cos(2t)\le1\)
From this, we may conclude that:
\(\displaystyle d_{\max}=2\) when \(\displaystyle \cos(2t)= -1\implies t=\dfrac{\pi}{2}\)
which is the diameter of a circle whose radius is 1 unit.
\(\displaystyle x(t)=\sin(2t)\)
\(\displaystyle y(t)=-\cos(2t)+1\)
For \(\displaystyle 0\le t<\pi\)
Hence, the chord length \(\displaystyle d\) will have a length whose square is:
\(\displaystyle d^2=x^2(t)+y^2(t)= \sin^2(2t)+\cos^2(2t)-2\cos(2t)+1= 2(1-\cos(2t))\)
We know:
\(\displaystyle -1\le\cos(2t)\le1\)
From this, we may conclude that:
\(\displaystyle d_{\max}=2\) when \(\displaystyle \cos(2t)= -1\implies t=\dfrac{\pi}{2}\)
which is the diameter of a circle whose radius is 1 unit.