Identities in algebra can sometimes be shown as area relationships. For example: The figure shown corresponds to the identity (a + b)2 = a2 + 2ab + b2. Do you see how? Draw figures corresponding to the identities (a + b)(a − b) = a2 − b2 and (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
Step-by-step solution
Idea: Area of a rectangle = length × breadth. If a shape is split into pieces, its area is the sum of the pieces’ areas; the two ways of counting give the two sides of the identity.
- Do you see how? The whole square has side a + b, so its area is (a + b)2. The lines split it into a square a × a, two rectangles a × b and a square b × b. Adding the parts: a2 + 2ab + b2.1 mark
- (a + b)(a − b) = a2 − b2: start with a square of side a and remove a corner square of side b. What is left has area a2 − b2. Cut it into P (a × (a − b)) and Q ((a − b) × b).1 mark
- Turn Q and place it beside P: together they form a rectangle of length a + b and breadth a − b. Same pieces, same area: (a + b)(a − b) = a2 − b2.1 mark
- (a + b + c)2: divide each side of a square of side a + b + c into parts a, b, c and draw lines across. The 9 pieces are three squares a2, b2, c2 on the diagonal and two rectangles each of ab, bc and ca. So (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.1 mark
Check: a = 6, b = 2: 36 − 4 = 32 = 8 × 4 ✓. a = 3, b = 2, c = 1: 36 = 9 + 4 + 1 + 12 + 4 + 6 ✓.
Answer to write in the exam
Square of side (a + b) = a2 + ab + ab + b2 ⇒ (a + b)2 = a2 + 2ab + b2
First figure: a × a square minus b × b corner = a2 − b2
Pieces a × (a − b) and b × (a − b) rearranged ⇒ rectangle (a + b) × (a − b)
∴ (a + b)(a − b) = a2 − b2
Second figure: square of side (a + b + c) = 3 × 3 grid: a2, b2, c2, 2ab, 2bc, 2ca
∴ (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Common mistakes that cost marks
- Drawing only one ab rectangle in (a + b)2; there are two.
- In the a2 − b2 figure, labelling the remaining strips with b instead of a − b.
- In the 3 × 3 grid, counting each of ab, bc, ca once instead of twice.
How this can come in the exam
A square of side (x + 3) is split into a square, two rectangles and a small square. The area of each rectangle is
- 3
- 3x
- 9
- x2
Show answer
(B) 3x
Each rectangle is x × 3 = 3x; total (x + 3)2 = x2 + 6x + 9.
Try one yourself
Use the area figure for (a + b)(a − b) to find 13 × 7 quickly.
Show answer
a = 10, b = 3: 102 − 32 = 91.
More questions like this
- An isosceles triangle has perimeter 40 cm; the equal sides are 15 cm each. Find the area of the triangle.
- An isosceles triangle has base 10 cm, and its area is 60 cm2. What are the lengths of the equal sides?
- The area of a right-angled triangle is 54 sq. cm. One of its legs has length 12 cm. Find its perimeter.
- The sides of a triangle are in the ratio 2: 3: 4, and its perimeter is 45 cm. Find its area.
- The sides of a triangle have lengths 7 cm, 24 cm, 25 cm. Find the area of the triangle in two different ways.