Concept explainers
To explain: Centre of gravity of a parallelogram.
Explanation of Solution
Given information
The figure is a parallelogram
Formula used:
Centre of gravity of figures is known as centroid:
Proof:
Let PQRSbe a parallelogram with the vertex P, Q, R and S
Let
Let
Let
Let
The vector PQ has the projections
The vector QR has the projections
The vector PS has the projections
The vector SR has the projections
The diagonal PR of this parallelogram represents the sum of vectors:
Let C be the intersection point of the diagonals PR and QS of the parallelogram
PQRS and the diagonals of a parallelogram bisect eachother at the intersection
The vector PC can be expressed by the two ways:
From the expression (1)
Hence, the x-coordinate of the point C is
Similarly, the y-coordinate of the point C is:
From the expression (2),
The vector PC has the x- and y- components
Hence, the x-coordinate of the point C is
Similarly, the y-coordinate of the point C
Since
This is exactly what has to be proved.
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