Difference between revisions of "Mass points"
Mathislife16 (talk | contribs) (I couldn't understand it from this article, so I researched it elsewhere and re-explained it.) |
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Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths must be inversely proportional to their weight (just like a balanced lever). Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever). | Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths must be inversely proportional to their weight (just like a balanced lever). Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever). | ||
| − | The way to systematically assign weights to the points involves first choosing a point for the entire figure to balance around. From there, WLOG a first weight can be assigned. From the first weight, others can be derived using a few simple rules. Any line passing this central point will balance the figure. If two points balance, the product of the mass and distance from a line of balance of one point will equal the product of the mass and distance from the same line of balance of the other point. | + | The way to systematically assign weights to the points involves first choosing a point for the entire figure to balance around. From there, WLOG a first weight can be assigned. From the first weight, others can be derived using a few simple rules. Any line passing this central point will balance the figure. If two points balance, the product of the mass and distance from a line of balance of one point will equal the product of the mass and distance from the same line of balance of the other point. If two points are balanced, the point on the balancing line used to balance them has a mass of the sum of the masses of the two points. |
For a full explanation with images, go to this link: http://old.mathtalentquest.org/images/Mass_Point.pdf | For a full explanation with images, go to this link: http://old.mathtalentquest.org/images/Mass_Point.pdf | ||
Revision as of 18:38, 28 February 2017
Mass points is a technique in Euclidean geometry that can greatly simplify the proofs of many theorems concerning polygons, and is helpful in solving complex geometry problems involving lengths. In essence, it involves using a local coordinate system to identify points by the ratios into which they divide line segments. Mass points are generalized by barycentric coordinates.
Mass point geometry involves systematically assigning 'weights' to points using ratios of lengths relating vertices, which can then be used to deduce other lengths, using the fact that the lengths must be inversely proportional to their weight (just like a balanced lever). Additionally, the point dividing the line has a mass equal to the sum of the weights on either end of the line (like the fulcrum of a lever).
The way to systematically assign weights to the points involves first choosing a point for the entire figure to balance around. From there, WLOG a first weight can be assigned. From the first weight, others can be derived using a few simple rules. Any line passing this central point will balance the figure. If two points balance, the product of the mass and distance from a line of balance of one point will equal the product of the mass and distance from the same line of balance of the other point. If two points are balanced, the point on the balancing line used to balance them has a mass of the sum of the masses of the two points.
For a full explanation with images, go to this link: http://old.mathtalentquest.org/images/Mass_Point.pdf
Examples
Consider a triangle 
with its three medians drawn, with the intersection points being 
corresponding to 
and 
respectively. Thus, if we label point 
with a weight of 
, 
must also have a weight of since and are equidistant from 
. By the same process, we find 
must also have a weight of 1. Now, since and both have a weight of , must have a weight of 
(as is true for 
and 
). Thus, if we label the centroid 
, we can deduce that 
is 
- the inverse ratio of their weights.
Problems
2013 AMC 10B Problems/Problem 16
2004 AMC 10B Problems/Problem 20
2009 AIME I Problems/Problem 5
2009 AIME I Problems/Problem 4
2001 AIME I Problems/Problem 7
External links
- http://mathcircle.berkeley.edu/archivedocs/2007_2008/lectures/0708lecturespdf/MassPointsBMC07.pdf
- http://mathcircle.berkeley.edu/archivedocs/1999_2000/lectures/9900lecturespdf/mpgeo.pdf
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