What is the union of the 4 half-planes determined by 2 intersecting lines.

a. the entire plane
b. the entire plane except for the two intersecting lines
c. the entire plane except for one of the lines
d. the entire plane except for the point of intersection
e. None of the Above

Did you make a sketch of 2 intersection straight lines?

What is meant by "union" ?

All the elements in the set: union

Its pretty much asking what is contained in 4 half-planes

To find the union of the four half-planes determined by two intersecting lines, you need to analyze the geometry and determine which areas are common to all four half-planes.

Here's how you can approach this problem:
1. Visualize the scenario: Draw two intersecting lines on a piece of paper or imagine them in a coordinate plane.
2. Understand the concept of half-planes: Each line divides the plane into two half-planes (one on each side of the line).
3. Identify the regions: Look at the areas formed by the intersecting lines. There are four regions in total: one to the left of both lines, one to the right of both lines, and two regions between the lines.
4. Determine the union: Now, consider the regions of overlap from the four half-planes. Note that the region to the left of both lines and the region to the right of both lines are common in all four half-planes. However, the regions between the lines are different in each half-plane.
5. Analyze the options:
a. The entire plane: Since the regions between the lines are not included in the union, option (a) is incorrect.
b. The entire plane except for the two intersecting lines: This option would include the regions between the lines, which are excluded from the union. Hence, option (b) is incorrect.
c. The entire plane except for one of the lines: This option would include the regions between the lines, which are also excluded from the union. Therefore, option (c) is incorrect.
d. The entire plane except for the point of intersection: This option is correct. The union of the four half-planes excludes the point of intersection, but includes all other regions.
e. None of the Above: This option is incorrect, as we have determined that option (d) is the correct answer.

Therefore, the answer is (d) the entire plane except for the point of intersection.