# For a linear triangular element with (xi, yi) as the coordinates of the ith node of the element the area=10units, the value of ∑γi from the standard relation αi+βiX+γiY=(2/3)*Area where X=∑xi, Y=∑yi is ___

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For a linear triangular element with (xi, yi) as the coordinates of the ith node of the element the area=10units, the value of ∑γi from the standard relation αi+βiX+γiY=(2/3)*Area where X=∑xi, Y=∑yi is ___

(a) 0

(b) 10

(c) 20

(d) 30

This question was posed to me at a job interview.

This interesting question is from Boundary Value Problems in section Single Variable Problems of Finite Element Method

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Best explanation: A linear triangular element has 3 nodes. With (xi, yi) as coordinates of ith node, the twice of area is given by determinant of the matrix$\begin{pmatrix}1&x1&y1\\1&x2&y2\\1&x3&y3\end{pmatrix}$ which equals to (x1y2−x2y1)+(x2y3−x3y2)+(x3y1−x1y3).Then from the standard relation we have ∑γi=−(x2−x3)−(x3−x1)−(x1 − x2)

=−x2+x3−x3+x1−x1+x2.

=0.

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