Sum and average length of chordsFrom a specific point A on a circle's circumference, am drawing various chords. a) what is the sum of length of such chords and b) what is the average length of such chords?Assuming a radius r, I'm trying to solve a) in the below manner:One specific chord length = 2 r cos ⁡ ( Θ )To get the sum, I'm integrating over − π / 2 t o π / 2 ( 2 r cos ⁡ ( Θ ) d Θ )But i don't think this is right. Please point me in the correct direction.

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Sum and average length of chordsFrom a specific point A on a circle&#039;s circumference, am drawing various chords. a) what is the sum of length of such chords and b) what is the average length of such chords?Assuming a radius r, I&#039;m trying to solve a) in the below manner:One specific chord length   =  2  r  cos  ⁡  (  Θ  )To get the sum, I&#039;m integrating over   −  π      /    2     t  o     π      /    2  (  2  r  cos  ⁡  (  Θ  )  d  Θ  )But i don&#039;t think this is right. Please point me in the correct direction.

Aedan Hatfield · Accepted Answer

Step 1There are many possible answers, depending on the probability distribution of the parameter that defines the other endpoint of the chord. That is a standard issue in problems of geometric probability.We give one interpretation, and derive the answer under that interpretation. Other interpretations will yield a different answer.Imagine that we select the other endpoint B by choosing   θ with uniform distribution in the interval   [  0  ,  2  π  ], and letting B be the point obtained by rotating the point A counterclockwise through the angle   θ.Step 2Then the length X of the chord AB is, by basic trigonometry,   2  r  sin  ⁡  (  θ      /    2  ). It follows that  E  (  X  )  =      ∫    0          2      π        2  r  sin  ⁡  (  θ      /    2  )      1          2      π          d  θ  .Integrate. We get  E  (  X  )  =            4      r        π    .

kaltEvallwsr · Accepted Answer

Step 1If you imagine many chords of a circle arranged in a stack, with appropriate lengths, it can be made into a circle. so, the total of all the chords will give us the area of the circle. This can be easily proved and imagined. For example, take a rectangle. Have you ever wondered why the area is l x b?? It is because we are adding the length breadth number of times.Step 2Hence applying the same logic to a circle, we can understand that if we add the lengths of all the chords in a circle we would get the circle&#039;s area.   π  ⋅      R    2  . The total number of chords would be equal to   2  ⋅  R as the length of the circle is   2  ⋅  R. Hence the average chord length   =  (  π  ⋅      R    2    )      /    (  2  ⋅  R  )  .  ⇒  π  ⋅  r      /    2

From a specific point A on a circle's circumference, am drawing various chords. a) what is the sum of length of such chords and b) what is the average length of such chords?

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