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?

Rosemary Chase

Rosemary Chase

Answered question

2022-11-15

Sum and average length of chords
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?
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.

Answer & Explanation

Aedan Hatfield

Aedan Hatfield

Beginner2022-11-16Added 16 answers

Step 1
There 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 2
Then 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

kaltEvallwsr

Beginner2022-11-17Added 8 answers

Step 1
If 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 2
Hence 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'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

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