The diameter is the longest cord in the circle

shahar

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How can I prove that diameter is the longest cord in the circle?
 
How can I prove that diameter is the longest cord in the circle?
If θ\theta is the measure of a central angle is a circle of radius R\mathit{R}
then the length of the subtended chord is c=2Rsin(θ2){\bf c}=2\mathit{R}\cdot\sin\left(\dfrac{\theta}{2}\right)
The length of a diameter is D=2RD=2\mathit{R}
The size of a cental angle determined by a diameter is π radians\pi\text{ radians}
 
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How can I prove that diameter is the longest cord in the circle?
There is a diameter, and here is a lovely little evidence I devised decades ago.

Make a circle O with any chord AB on it. Draw a line segment through the centre of the chord from one endpoint, say A. That is, draw a circle. View the diagram. Draw a radius from centre O to centre B.
According to the triangle inequality, AB AO + OB = r + r =2r = d. As a result, every chord that is NOT a diameter is less than a diameter. As a result, the largest chord is a diameter.
math.PNG
 
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