NCERT Solutions for Class 9th Maths Chapter 4 Think and Reflect — More Identities
Book page 76 Updated on2026-09-08
Q1.
Label the squares and rectangles in Fig. 4.4 so that it represents the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
Answer
The side of the big square is split into a, b and c both horizontally and vertically, so the nine pieces have areas (row height) × (column width):
A square of side a + b + c cut by both splittings. The three squares on the diagonal give a², b² and c²; the six rectangles pair up into 2ab, 2bc and 2ca.
column a
column b
column c
row a
a²
ab
ac
row b
ab
b²
bc
row c
ac
bc
c²
Total area = a² + b² + c² + (ab + ab) + (bc + bc) + (ac + ac)
= a² + b² + c² + 2ab + 2bc + 2ca
and that same area is (a + b + c)², being a square of side a + b + c.
Why the rectangles come in pairs: the piece in row a, column b and the piece in row b, column a are different rectangles in the picture but have the same area, ab — one is a tall ab, the other a wide ab. The grid is symmetric about its main diagonal, so every off-diagonal area appears exactly twice. That symmetry is the geometric reason each cross term carries a factor 2, and it is why the identity has 2ab, 2bc, 2ca and not ab, bc, ca.
Tip: The same picture in n strips gives (a₁ + a₂ + … + aₙ)² = sum of all aᵢ² + twice the sum of all products aᵢaⱼ with i < j. Three letters is just the first interesting case.