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).
We will discuss today the three different situations that exist in MPG problems, explore them and then slowly transit to some composite problems where in MPG is used as a sub part to the whole solution which will show us its true utility in geometry.
Case of Concurrent Cevians
A Cevian is any line segment in a triangle with one endpoint on a vertex of the triangle and the other endpoint on the opposite side. Medians, altitudes, and angle bisectors are special cases of cevians.
Assign the vertex common to sides for which ratio split is given and set it as the LCM of two ratio points away from it so that all mass points assigned remain whole numbers.
Case of Splitting Masses
The lone vertex on one side of transversal (DE here) has split masses. That is to say, it is a common point to two see-saws having different second ends. Hence the mass points add up from both sides or are considered as split masses to balance both sides of the see saw. First assignment is made at any of the two vertices(A,B) away from the lone vertex(C). The assignment made so that we get whole numbers at each point. So to achieve that we assign LCM(3,5) = 15 to the point A and then following simple assignment of inverse ratio mass points and added sum at the fulcrum point.
Lone vertex,C gets 10m from the AC see saw and 9m from the BC see saw achieving a total of 19m to balance all see saws.
Case of Multiple SystemsIn case we have two central points on a single see saw(any cevian) we have on our hands a case of multiple systems. In such a case we remove one line segment(containing one of the central points) at a time and solve the system assuming that line segment is absent. It would leave us a system which one of the above 2 cases and we can easily solve that. Similarly we do it after removing the other line segment with the other central point. Then use the ratios obtained from two separate systems to solve our problem by simple manipulation of ratios calculated.
This rounds up the basics of mass point geometry which would allow you to solve basic questions based on the concept. In the next article of this series we'll look at some complex and composite problems that make use of MPG at some stage to reach the final answer.
Rate,Like and Share the posts if you like the content on Lagom.
Cheers!
AS
good theorem to learn
ReplyDelete