High School: Geometry
Expressing Geometric Properties with Equations HSG-GPE.B.6
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Ross and Rachel. Rachel and Ross. Every Friday morning, we gathered around the break room coffee maker to discuss what it would take to split them up. Just like the 18-page wedge that finally came between the two old friends, we can also identify the coordinates of a partition point that splits up a directed line segment.
The three main methods that students can use to find the partition point are as distinct from each other as sitcom character archetypes. Students should be able to use the midpoint formula (if the ratio of the parts is 1:1), the section formula, and the distance formula.
If we're being honest, though, the midpoint formula is just a special case of the section formula. But it comes first in the list because it's the easiest (unlike the game show Pyramid). Just find the averages of the x and y coordinates to find the midpoint, which gives the partition point these coordinates.
The section formula is just a fancier version of the midpoint formula. 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 students should plug the numbers into the section formula to find the coordinates of P.
Essentially, the midpoint formula is to finding averages as the section formula is to finding weighted averages. Given the endpoints of a line segment, students should be able to use both formulas to find the midpoint M and the partition point P at a specified ratio.
Students should also be able to determine the ratio of a partition using the distance formula. They need to remember that we are talking about directed line segments here. So it does matter on which side of the partition point the bigger segment lies. Remind them to keep track of which segments they're looking at, and hopefully they won't get mad at you. It's not as though you called them boring or anything.
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