Mass Point Geometry

Well, as I am preparing for CAT 2012, so i thought it would be nice to share some out of the box things which i learn at my coaching classes. This might help you somewhere, and will be saved as an archive for my reference also. There is something very special kind of geometry I learned last week, and I am now exploring all the application part. The concept is very simple, and can be applied to any problem. Here is how I can explain it-

The concept-
It uses very simple principle of physics, which we have often observed in our daily life, see the figure below

Consider AB as line segment and C as some point in between, if the length of AC = 2 unit and length of BC = 8 unit, then the amount of weights need to balance will be 8 units on the side AC and 2 units on the side BC, as shown in figure.
Now this as obvious concept can be used in solving few problems relating to geometry, taking an example I can explain how..

An Example-
Question : – Consider the case below here are of small triangles contained in big triangle are known, now we have to find the area of whole triangle, I bet you can’t do this question in less than 10 minutes if you don’t know this method. Give it a try if you want to, before reading the solution.

Answer : – As we know base of triangle is in the same ratio as the areas are, when a line cuts triangle, hence the ratio of base will be same as ratio of areas.. Therefore, ratio of base-

 Now, using mass geometry i can complete this triangle as shown below-

Here, numbers written are nothing but ratios of sides, we know the area and ratio of sides. Therefore, we can find unknown area of quadrilateral. Picture below shows the actual areas in squares and ratios of sides. Using this we can say that, ratio of area of triangle ABE and triangle BCE = (4+x)/(8+13) = 7/1, where x is area of quadrilateral.

Therefore, 4+x = 7(8+13) = 21 x 7 = 147 => x = 143
Therefore, area of triangle ABC = 143 + 4 + 8 + 13 = 168
Please let me know through comments if you have any doubts. 🙂

10 thoughts on “Mass Point Geometry”

  1. Hi Prateek,
    Please help me with this using Mass point Geometry.

    In a triangle ABC, let E be a point on AB such that AE : EB = 1 : 3, D is a point on BC such that BD : DC = 4 : 1, and F is a point on ED joined such that EF : FD = 5 : 1. Show that
    AG : GC = 4 : 1 and BF : FG = 17 : 7.

    How to balance this problem to arrive at the answer ?

  2. hi prateek
    i have a doubt here …
    i didn't get this step
    Answer : – As we know base of triangle is in the same ratio as the areas are, when a line cuts triangle, hence the ratio of base will be same as ratio of areas.. Therefore, ratio of base-
    and how you simplified it later , could you elobarate it a bit ?

  3. Area of triangle are in ratio, 4:8, with the same base, Hence base will be in ratio 1:2, therefore 7:14(=1:2)
    I divided it in terms of 7:14 because, 8+13 =21. Now we have to divide this 21 in ratio 1:2. Hence, 7:14

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