Class 9 Exam > Class 9 Notes > Lab Manual: Verify the Algebraic Identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Lab Manual: Verify the Algebraic Identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca - Class 9 PDF Download
Objective
To verify the algebraic identity (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
Materials Required
- Hardboard
- Coloured papers
- Adhesive
- White paper
- Scissors
- Geometry Box
Prerequisite Knowledge
- Square and its area.
- Rectangle and its area.
Theory
- For square and its area refer to Activity 3.
- For rectangle and its area refer to Activity 3.
Procedure
- Take a hardboard of suitable size and paste a white paper on it.
- From a coloured paper, cut out a square of side a units, (see Fig. 6.1)
- Further, cut out a square of sided units (b < a)from another coloured paper, (see Fig. 6.2)
- Also, cut out a square of sidec units (c < b)from different coloured paper.(see Fig. 6.3)
- Cut out two rectangles of dimensions b x a from different coloured paper, (see Fig. 6.4)
- Also, cut out two rectangles of dimensions c x b from different coloured paper, (see Fig. 6.5)
- Now further, cut out two rectangles of dimensions c x a from another coloured paper, (see Fig. 6.6)
- Paste the squares and rectangles on the hardboard as shown in Fig. 6.7.
Demonstration
From Fig. 6.7, it is clear that from the arrangement of sqaures and rectangle, square PQRS of side (a + b+c) units is obtained.
Area of square PQRS = (a + b +c)2 [∴ area of square = (side)²] … (i)
Also, area of square PQRS = Sum of the areas of all the squares and rectangles, which are used to make the square PQRS = a2 + b2 + c2 + ab + ab + bc + bc + ca + ca = (a2 + b2 + c2 + 2ab + 2bc + 2ca) …(ii)
From Eqs. (i) and (ii), we have (a + b + c)2 = (a2 + b2 + c2 + 2ab + 2bc + 2ca) Here, area is in square units.
Observation
On actual measurement, we get
a = …….. , b = …….. , c = …….. ,
So, a2 = …….. , b2 = …….. , c2 = …….. ,
ab = …….. , bc = …….. , ca = …….. ,
2ab = …….. , 2bc = …….. , 2ca = …….. ,
a + b + c = …….. ,
and (a + b + c)2 = …….. ,
Hence, (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Result
Identity (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca has been verified.
Application
This identity may be used for
- calculating the square of a number which can be expressed as a sum of three convenient numbers.
- simplification and factorisation of algebraic expressions.
FAQs on Lab Manual: Verify the Algebraic Identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca - Class 9
| 1. What is the algebraic identity (a + b + c)²? |  |
| 2. How can I verify the algebraic identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca? | |
Ans. To verify the algebraic identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca, you need to expand both sides of the equation. Start by squaring the binomial (a + b + c) and then simplify both sides to see if they are equal.
| 3. Can this algebraic identity be used for any values of a, b, and c? | |
Ans. Yes, this algebraic identity can be used for any values of a, b, and c. It is a general formula that holds true for all real numbers.
| 4. Is the algebraic identity (a + b + c)² commutative? | |
Ans. No, the algebraic identity (a + b + c)² is not commutative. This means that if you change the order of the terms, the result will be different. For example, (a + b + c)² is not the same as (b + a + c)².
| 5. Can you provide an example of using this algebraic identity in a problem? | |
Ans. Sure! Let's say we have a = 2, b = 3, and c = 4. We can substitute these values into the algebraic identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. After simplifying both sides, we should get the same result, which confirms the validity of the identity.
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