Exercise 11.5

1. Verify the identity a+b2 = a2 + 2ab + b2 geometrically by taking

(i) a = 2 units, b = 4 units

Solution:

We can represent a+b2 as the area of a square with side length (a + b). In this case, the side length is 2 + 4 = units.

Imagine a square. Divide it into four regions.

A square with side 'a' (2 units): Area = a2 = 22 = square units.

A rectangle with sides 'a' and 'b' (2 and 4 units): Area = ab = 2 × 4 = square units.

A rectangle with sides 'b' and 'a' (4 and 2 units): Area = ba = 4 × 2 = square units.

A square with side 'b' (4 units): Area = b2 = 42 = square units.

The total area of the large square is a+b2 = 62 = square units.

The sum of the areas of the four regions is a2 + 2ab + b2 = 4 + 8 + 8 + 16 = square units.

Since both areas are , the identity ``(a + b)^2=a^2+2ab+b^2` is verified geometrically for a = 2 and b = 4.

(ii) a = 3 units, b = 1 unit

Solution:

The side length of the square is 3 + 1 = units. The total area is a+b2 = 42 = square units.

Imagine a square. Divide it into four regions.

A square with side 'a' (3 units): Area = a2 = 323 = square units.

A rectangle with sides 'a' and 'b' (3 and 1 units): Area = ab = 3 × 1 = square units.

A rectangle with sides 'b' and 'a' (1 and 3 units): Area = ba = 1 × 3 = square units.

A square with side 'b' (1 unit): Area = b2 = 12 = square unit.

The sum of the areas is 9 + 3 + 3 + 1 = square units.

The identity is verified geometrically for a = and b = .

(iii) a = 5 units, b = 2 units

Solution:

The side length of the square is 5 + 2 = units. The total area is a+b2 = 72 = square units.

Imagine a square. Divide it into four regions.

A square with side 'a' (5 units): Area = a2 = 52 = square units.

A rectangle with sides 'a' and 'b' (5 and 2 units): Area = ab = 5 × 2 = square units.

A rectangle with sides 'b' and 'a' (2 and 5 units): Area = ba = 2 × 5 = square units.

A square with side 'b' (2 units): Area = b2 = 22 = square units.

The sum of the areas is 25 + 10 + 10 + 4 = square units.

The identity is verified geometrically for a = 5 and b = 2.

2. Verify the identity (a – b)^2 = a^2 – 2ab + b^2 geometrically by taking

(i) a = 3 units, b = 1 unit

Solution:

a−b2 represents the area of a square with side length (a - b). In this case, it's (3 - 1) = units. The area is 22 = square units.

Imagine a square with side 'a' (3 units). Remove a rectangle of width 'b' (1 unit) from one side and another rectangle of width 'b' from the adjacent side. The remaining area is a−b2.

Start with a square of side 'a' (3 units): Area = a2 = 32 = square units.

Remove a rectangle of sides 'a' and 'b' (3 and 1 units): Area = ab = 3 × 1 = square units.

Remove another rectangle of sides 'b' and 'a' (1 and 3 units): Area = ba = 1 × 3 = square units.

The area removed twice (b2) needs to be added back once as it was subtracted twice: Area = b2 = 12 = square unit.

The remaining area is a2 - 2ab + b2 = - 2() + = square units.

Since both areas are , the identity is verified geometrically for a = 3 and b = 1.

(ii) a = 5 units, b = 2 units

Solution:

a−b2 represents the area of a square with side length (5-2) = units. The area is 32 = square units.

Imagine a square of side 'a' (5 units). Remove a rectangle of width 'b' (2 units) from one side and another rectangle of width 'b' from the adjacent side. The remaining area is a−b2.

Start with a square of side 'a' (5 units): Area = a2 = 52 = square units.

Remove a rectangle of sides 'a' and 'b' (5 and 2 units): Area = ab = 5 × 2 = square units.

Remove another rectangle of sides 'b' and 'a' (2 and 5 units): Area = ba = 2 × 5 = square units.

The area removed twice b2 needs to be added back once: Area = b2 = 22 = square unit.

The remaining area is a2 - 2ab + b2 = 25 - 10 - 10 + 4 = square units.

The identity is verified geometrically for a = 5 and b = 2.

3. Verify the identity (a + b)(a – b) = a2 – b2 geometrically by taking

(i) a = 3 units, b = 2 units

Solution:

(a + b)(a - b) represents the area of a rectangle.

(a + b)(a - b) = (3 + 2)(3 - 2) = 5 × 1 = square units.

a2 - b2 = 323 - 222 = - = square units.

The identity is verified geometrically for a = 3 and b = 2.

(ii) a = 2 units, b = 1 unit

Solution:

(a + b)(a - b) = (2 + 1)(2 - 1) = × = square units.

a2 - b2 = 22 - 12 = - = square units.

The identity is verified geometrically for a = 2 and b = 1.