Dividing a Line Segment into Equal Parts
Date: 11/22/2004 at 08:23:05 From: Iain Subject: Determining coordinates of equal parts of a line segment I have two endpoints of a line segment with coordinates A(2, 7) and B(-4, -2). I am looking for the coordinates of the points that divide AB into 3 equal parts. If drawn on a graph, the points that divide AB are shown clearly. But what if the coordinates of AB do not allow such convenient results?
Date: 11/22/2004 at 09:41:20 From: Doctor Barrus Subject: Re: Determining coordinates of equal parts of a line segment Hi, Iain! Let's look at a picture of a line segment: O A(2, 7) / / / / / O B(-4, -2) Say we want to split segment AB into 3 equal parts, like this: O A(2, 7) / . D(?, ?) / . C(?, ?) / O B(-4, -2) If I understand you correctly, you want to find out the coordinates of points C and D, right? Well, in order to help you understand my answer, I'm going to add a little bit more to my drawing. First I'll draw a vertical line through B and a horizontal line through A, which will form a triangle: O A(2, 7) /| D . | / | C. | / | O-----O E(2,-2) B(-4, -2) Notice that since E is directly below A, its x-coordinate will be 2, the same as A's. Since E is directly to the right of B, its y- coordinate will be -2, the same as y's. Now I'm going to mark points on segment BE directly below points C and D, and I'm going to mark points on segment AE directly to the right of C and D: O A /| D . . H / | C. . I / | O-.-.-O E B F G Now the x-coordinate of point C is the same as the x-coordinate of point F, right? So I'm going to try to find the x-coordinate of point F, and when I do, the answer will also be the x-coordinate of point C. Let's just look at that bottom side of the triangle, segment BE: -4 2 O-------O-------O-------O B F G E The x-coordinate of B is -4, and the x-coordinate of E is 2. So the distance between B and E is 6, because 2 - (-4) = 6. -4 2 O-------O-------O-------O B F G E <--------- 6 ---------> Now F is 1/3 of the way from B to E. Since the total distance from B to E is 6, the distance from B to F is (1/3)*6 = 2 (Here * means multiplication). So F is 2 units away from B, and G is 2 units away from F. So we get the picture -4 -2 0 2 O-------O-------O-------O B F G E <- 2 -> <- 2 -> <- 2 -> So the x-coordinate of F is -2, and the x-coordinate of G is 0. If we look back the triangle we drew, we see that C also has to have x- coordinate -2, and D has to have x-coordinate 0. O A /| D . . H / | C. . I / | O-.-.-O E B F G Now look at the segment AE with its y-coordinates: 7 O A | | O H | | O I | | -2 O E Can you find out what the y-coordinates for H and I should be? The answers should tell you what the y-coordinates for D and C should be. I hope this has helped. If you'd like a little more explanation, please write us back with your questions. Good luck! - Doctor Barrus, The Math Forum http://mathforum.org/dr.math/
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