Observe the two rows of figures below. They represent an algebraic identity. Try to identify it.       [Page No. 78]

Consider the identity

$$(a + b + c)^2 + (a + b - c)^2 + (a - b + c)^2 + (a - b - c)^2 = 4a^2 + 4b^2 + 4c^2.$$

On expanding the four squares on the left, the cross-terms $2ab, 2bc$, and $2ca$ appear equally often with positive and negative signs. Hence, all the cross-terms cancel out.

Only the square terms remain, giving $4a^2 + 4b^2 + 4c^2$.

Since, $4a^2 + 4b^2 + 4c^2 = (2a)^2 + (2b)^2 + (2c)^2$, the sum of the areas of the four squares in the top row is equal to the sum of the areas of the three squares with sides $2a$, $2b$, and $2c$ in the bottom row. This provides a neat geometric interpretation of the identity.