1. Midpoint Formula & Partitions
Topic/Objective: Partitions Essential Question: How do we partition a Line Segment in the coordinate plane?
Vocabulary: Midpoint, Partition, Ratio, Proportion

2. A line segment is a part of a line that is noted by two endpoints, ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 ).
You can partition or divide a line segment in lots of different ways.
A popular partition is dividing a line segment into two equal parts. The point on the segment where this division occurs is called the midpoint.
Introduction

3. Example of Line Segment with partitions
We will learn how to coalculate the ratio later in the slide show…

4. Steps to Finding the Midpoint of a Line Segment
Determine the endpoints of the line segment, (x1, y1) and (x2, y2).
Write the Midpoint Formula.
Substitute the values of (x1, y1) and (x2, y2) into the midpoint formula:
Simplify.

5. Practice This…
Calculate the midpoint of the line segment with endpoints (-2,1) and (4, 10).

6. The coordinates of an endpoint Can be calculated from the other endpoint and the midpoint

7. Watch this:
A line segment has an endpoint at (6, 0) and a midpoint at (3,1).
Determine the coordinates of the other endpoint.

8. Practice this…
A line segment has an endpoint (2, -2) and a midpoint at (3, 0).
Determine the coordinates of the other endpoint.

9. Steps to Partitioning a Line Segment with any given fraction or ratio
*If given a ratio, convert it to a fraction.
Draw the line segment on a coordinate plane.
Calculate the difference between the x-values: |x2 – x1|. Multiply your answer to the given fraction. Move this many units to the left or the right of your endpoint on your graph.
Calculate the difference between the y-values: |y2 – y1|. Multiply your answer to the given fraction. Move this many units up or down from where you left off at on the graph at the end of step 2.
State the point where the segment is partitioned.

10. Guided Practice Determine the point that is 1 4 the distance from the
endpoint (-3, 7) of the
segment with endpoints
(-3, 7) and (5, -9).
Draw the segment on
a coordinate plane.
6.2.1: Midpoints and Other Points on Line Segments

11. Guided Practice 2. Calculate the difference between the
x-values |x2 – x1|. _____________
Multiply this answer by the fraction.
________ ∙ 𝟏 𝟒 = ______
Move this many units to the left or
right of your endpoint (-3, 7)
on your graph.
6.2.1: Midpoints and Other Points on Line Segments

12. Guided Practice 3. Calculate the difference between the
y-values |y2 – y1|. _____________
Multiply this answer by the fraction.
________ ∙ 𝟏 𝟒 = ______
Move this many units up or down
from where you left off on the graph
from step 2.
4. State the point that is 𝟏 𝟒 the distance
from the endpoint (-3, 7): _________
6.2.1: Midpoints and Other Points on Line Segments

13. Independent Practice Determine the point that
is the distance from the
endpoint (4, 1) of the
segment with endpoints
(4, 1) and (10, 10).
6.2.1: Midpoints and Other Points on Line Segments

14. Summary
To find the point P that divides a segment AB into a particular ratio, determine the ratio k by writing the numerator over the sum of the numerator and the denominator of the given ratio. Next, find the rise and the run (slope) of the line. Finally, add (k • the run) to the x- coordinate of A and add (k • the rise) to the y-coordinate of A. This process is summarized with the following formula.