Chapter 9.4, Problem 49HP

To determine

To find out the total number of line that can be drawn so that each line passes through exactly 2 of the given 5 points

Answer to Problem 49HP

There are total 4 lines that can be drawn so that each line passes through exactly 2 of these 5 points.

Explanation of Solution

Given information:

Five distinct points lie in a plane such that 3 of the points are on the line $l$ and 3 of the points are on a different line $m$

Since one point is common to both lines, so 4 points are left through which the line can be drawn because any point with a line connecting to it to the common point will have to go through another point already on that line. Therefore, 4 points are left that can be connected to any other of these 4 points, except for one since it already lies on the same line with common point. Therefore, 2 of the points are eliminated that each point connect to, with 2 left. Once the point are connected the reverse of that line is the same line, so once a point is connected to another it leaves only one line that can be drawn from that point to a different one.

Therefore, there are total 4 lines that can be drawn so that each line passes through exactly 2 of these 5 points.