how does the no. of regions increase when another chord is added to the circle?
When another chord is added to a circle, the number of regions within the circle's interior increases. To understand how this happens, we need to consider the relationship between chords and regions.
The key concept here is that each chord divides the interior of a circle into two regions: one on each side of the chord. So, when you add a chord to a circle, it creates a new region between the existing chords.
To visualize this, imagine drawing a circle on a piece of paper. Then, draw a chord anywhere inside the circle. You will notice that it divides the inside of the circle into two separate regions. Now, draw another chord that intersects the first one. This new chord will create two more regions, one between the first chord and the outer boundary of the circle, and another between the second chord and the outer boundary of the circle.
As you continue to add more chords, each chord will intersect all the previously added chords, creating new regions each time. The number of new regions formed by adding a new chord is equal to the number of existing chords intersected by the new chord plus one.
To find the total number of regions when you have a certain number of chords and no overlapping occurs, you can use a formula:
Number of Regions = (Number of Chords * (Number of Chords + 1) / 2) + 1.
For example, if you have 3 chords in a circle, you can calculate the number of regions as follows:
Number of Regions = (3 * (3 + 1) / 2) + 1 = (3 * 4 / 2) + 1 = 6 + 1 = 7.
So, with 3 chords, the circle will be divided into 7 regions.
In summary, each chord added to a circle cuts the interior into two regions, resulting in an increase in the total number of regions formed by the chords.