Partitioning a Directed Line Segment

Do Now What is the midpoint formula?

Weighted Average A weighted average is a kind of arithmetic mean in which some elements carry more importance (weight) than others. Ex.: Suppose that homework counts 10%, quizzes 20%, and tests 70%. If Harry has a homework grade of 92, a quiz grade of 68, and a test grade of 81, then what is his overall grade? (10*92 + 20*68 + 70*81)/100 = 79.5

Partitioning a Directed Line Segment A midpoint partitions a segment into two smaller segments in the ratio __:__. If a line segment has endpoints (x1, y1) and (x2, y2), and a partition point P will separate the line segment into a ratio of m:n, then use this formula to find P: Directed line segment means… This is a weighted average. Order of the endpoints matters.

Ex. 1 Line segment AB has endpoints (7, 2) and (4, 6). What are the coordinates of the point that divides AB in the ratio of 2:3? ((2*4 + 3*7)/5, (2*6 + 3*2)/5) = (5.8, 3.6)

Ex. 2 Line segment CD has endpoints (-3, 8) and (1, -5). What are the coordinates of the point that divides CD in the ratio of 3:7? ((3*1 + 7*-3)/10, (3*-5 + 7*8)/10) (-1.8, 4.1)

Ex. 3 Line segment EF has endpoints (5.5, 2.25) and (6.5, -1.75). Find the coordinates of point P, that lies 5/8 of the way on the directed line segment EF? 5/8 means there are 8 parts, and 5 of those parts are on one side of the point. So 3 of them are on the other side of the point. So the ratio is 5:3. ((5*6.5 + 3*5.5)/8, (5*-1.75 + 3*2.25)/8) (6.125, -0.25)

Ex. 4 Line segment GH has endpoints with coordinates (-4, 11) and (8, -1). Find two possible locations for a point that divides the segment into two parts with lengths in a ration 1:3. By asking for two points, we’re being asked to consider both “directions.” ((1*8 + 3*-4)/4, (1*-1 + 3*11)/8) = (-1, 4) Reversing the direction: ((1*-4 + 3*8)/4, (1*11 + 3*-1)/8) = (5, 1)