Given 39 points, how many lines can be drawn that contain two of the 39 points?

Question

im supposed to know how many lines 39 points make

Answer 1

The answer depends on how the points are located relative to each other. If all 39 of them are colinear (i.e., already on a straight line), we can draw only one line through them. This is is the minimum.

 

Now for the maximum number. Suppose no three of them are colinear. Then they form the corners of a 39-gon. Pick one of the points. You can draw 38 lines to the other points. They all form edges in the 39-gon. Now pick the nearest neighbor to the first point. You can draw an additional 37 lines to the remaining points. Repeat this until you're at the 38th point: there is one line left you can draw, the one to the 39th point. Altogether, you have drawn 38+37+36+....+2+1 lines, for a total of 741 lines. As a shortcut, you can use the formula

∑_m=1^n-1 m = n(n-1)/2, with n=39.

To see what's going on graphically, you can use a pentagon (n=5) instead of a 39-gon. You can draw 5*4/2=10 edges in a pentagon.

So you can draw at least one line and at most 741 lines, depending of how many of your 39 points are colinear.

Answer 2

Pick a point and call it P0; you can draw a line through it and each of the other 38 unnamed points. That's 38 lines.

 

Pick another point, not P0, and call it P1; you can draw a line through it and each of the other 37 unnamed points. That's 37 more lines.

 

Pick another point, not already named, and call it P2; you can draw a line through it and each of the other 36 unnamed points. That's 36 more lines.

 

Continue the pattern through naming P37; you can draw a line through it and the last unnamed point (which you can name P38). That's 1 more line.

 

So the total number of lines is

S = 38 + 37 + ... +   2 +   1, or

S =   1 +  2  + ... + 37 + 38.

 

Adding the two: 2S = 39 + 39 + ... + 39 + 39,

2S = 38 * 39. Dividing by 2: S = 19 * 39, or

S = (20 - 1)(40 - 1) = 801 - 60 = 741.

 

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This is a variation of the "handshake" problem: As each person enters a room s/he shakes hands with everyone in the room once. After the 39th person has entered and shaken hands, how many handshakes have occurred?