For the questions in this activity, use the coordinate grid if it is helpful to you.
What is the midpoint of the segment connecting and ?
What is the midpoint of the segment connecting and ?
What is the midpoint of the segment connecting and ?
12.2
Activity
Part Way: Segment
Point has coordinates . Point has coordinates .
Find the point that partitions segment in a ratio.
Calculate .
What do you notice about your answers to the first 2 questions?
For 2 new points and , write an expression for the point that partitions segment in a ratio.
12.3
Activity
Part Way: Quadrilateral
Here is quadrilateral .
Find the point that partitions segment in a ratio. Label it .
Find the point that partitions segment in a ratio. Label it .
Find the point that partitions segment in a ratio. Label it .
Is a dilation of ? Justify your answer.
Student Lesson Summary
To find the midpoint of a line segment, we can average the coordinates of the endpoints. For example, to find the midpoint of the segment from to , average the coordinates of and : . Another way to write what we just did is or .
Now, let’s find the point that is of the way from to . In other words, we’ll find point so that segments and are in a ratio.
In the horizontal direction, segment stretches from to . The distance from 0 to 6 is 6 units, so we calculate of 6 to get 4. Point will be 4 horizontal units away from , which means an -coordinate of 4.
In the vertical direction, segment stretches from to . The distance from 4 to 7 is 3 units, so we can calculate of 3 to get 2. Point must be 2 vertical units away from , which means a -coordinate of 6.
It is possible to do this all at once by saying . This is called a weighted average. Instead of finding the point in the middle, we want to find a point closer to than to . So we give point more weight—it has a coefficient of rather than as in the midpoint calculation. To calculate , substitute and evaluate.
Either way, we found that the coordinates of are .
