1. Partitioning a Segment CC Standard G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Objective: Students will be able to find the ratio of a partitioned line segment. Inquiry Question: How do you find the point on a directed line segment that partitions the segment in a given ratio? What does partition mean? To break apart into pieces! Ratio: A comparison of 2 quantities.

2. Point P divides in the ratio 3 to 1. 1.What does this mean? 2. Do you expect point P to be closer to A or closer to B? Why? 3. How does the slope of AP compare with the slope of the AB? Why?

3. Slope Formula The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is as follows: y2 – y1y2 – y1x2 – x1x2 – x1y2 – y1y2 – y1x2 – x1x2 – x1 m =

4. Ex. Find the slope of the line that passes through (–2, –3) and (4, 6). Let (x 1, y 1 ) be (–2, –3) and (x 2, y 2 ) be (4, 6). 6 – (–3) 4 – (–2) Substitute 6 for y 2, –3 for y 1, 4 for x 2, and –2 for x 1. = y 2 – y 1 x 2 – x 1 9 6 = 3 2 = The slope of the line that passes through (–2, –3) and (4, 6) is. 3 2 *** Always reduce your fractions****

5. Find the coordinate of point P that lies along the directed line segment from A(3, 4) to B(6, 10) and partitions the segment in the ratio of 3 to 2. Directed Line Segment: Tells the direction in which from which point to start and end. In this case, from Point A to Point B So point A must be labeled A and B What does that tell you about the distance AP and PB in relation to AB? Example 1 :

6. Partitioning a segment 1.Label your points and Note: since it is a directed segment, order does matter. 3. Find the rise and run for the segment (order does matter) Rise: Run: 4. To find the partitioning point: x – coordinate: x 1 -coordinate + run(% in fraction form) y - coordinate: y 1 -coordinate + rise(% in fraction form) 2. Convert the ratio into a percent (keep as a fraction) a:b Percent ratio (%)

7. What is the ratio that segment AB is partitioned into a point by point P? 1.Focus on our y-intercepts. 2.AP: 4 - 8 = -4 PB: 8-10 = -2 *Our ratio has to be positive, so we use absolute value {-4} = 4, and {-2} = 2 3. The ratio that segment AB is partitioned into a point by point P is 4:2 or 2:1 (5,8)

8. Example: What is the ratio that segment AB is partitioned into a point by point P? B (4,6) A (-5,1) P (0,4)