1. Partitioning a Directed Line Segment

2. Do Now
What is the midpoint formula?

3. 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* * *81)/100 = 79.5

4. 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.

5. 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)

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)

7. 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* *5.5)/8, (5* *2.25)/8)
(6.125, -0.25)

8. 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)