In how many regions (parts) do three lines in general position divide a plane?
2 Parts.
In general, 7 parts. Draw a triangle with sides extended and this will be clear.
To determine the number of regions that three lines in general position divide a plane into, we can follow these steps:
1. Start with the empty plane as the first region.
2. Each line that is added to the plane will intersect the previously existing lines in a point. These points of intersection create new regions.
3. For every new line that intersects the existing lines, there will be a new point of intersection. This new point will create additional regions by splitting some of the existing regions.
4. The number of additional regions created by a new line is equal to the number of existing regions that are split by that line.
5. Therefore, to count the total number of regions, we need to determine the number of intersections caused by each line and add the total count of new regions created by each intersection.
The formula to calculate the number of regions produced by n lines in general position is given by:
R(n) = R(n-1) + n
For three lines, the formula will be:
R(3) = R(2) + 3
So, to find the number of regions that three lines in general position divide a plane into, we need to calculate R(3):
R(1) = 1 (empty plane with no lines)
R(2) = R(1) + 2 = 1 + 2 = 3 (one line divides the plane into two regions)
R(3) = R(2) + 3 = 3 + 3 = 6
Therefore, three lines in general position divide a plane into 6 regions.