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1) A circle has 1500 chords drawn in it so that the maximum number of non-overlapping regions is obtained. Find the difference between the number of points of intersection of the chords and the number of polygons formed by the chords inside the circle.

User Dierdre
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1 Answer

2 votes

Answer: 1,121,252

Explanation:

First, we need to find the number of points of intersection of the chords inside the circle. To do this, we can use the formula for the number of intersection points in a circle formed by n chords, which is given by:

Number of intersection points = n(n - 1)/2

In this case, n = 1500, so:

Number of intersection points = 1500(1500 - 1)/2

Number of intersection points = 1500 * 1499/2

Number of intersection points = 750 * 1499

Number of intersection points = 1,124,250

Now, let's find the number of polygons formed by the chords inside the circle. When n chords intersect inside a circle, they form (n - 2) triangles and n non-triangle polygons. In this case, n = 1500, so:

Number of triangles = (1500 - 2) = 1498

Number of non-triangle polygons = 1500

Now, calculate the total number of polygons:

Total number of polygons = Number of triangles + Number of non-triangle polygons

Total number of polygons = 1498 + 1500

Total number of polygons = 2998

The difference between the number of points of intersection of the chords and the number of polygons formed by the chords inside the circle is:

Difference = Number of intersection points - Total number of polygons

Difference = 1,124,250 - 2998

Difference = 1,121,252

So, the difference between the number of points of intersection of the chords and the number of polygons formed by the chords inside the circle is 1,121,252.

User Maxbellec
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