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In Class 9 Mathematics, under the chapter Exploring Algebraic Identities, students use geometric diagrams and area-based representations to understand algebraic identities visually. They relate squares and rectangles formed from algebraic expressions to expansions such as binomial products, helping them see why both sides of an identity are equal rather than merely memorising formulas.
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Easy · Level 73 · difference of squares,visual models,area of squares,algebraic identities,class 9 mathematicsView options
77
81
79
85
Easy · Level 73 · algebraic identities, visual models, area model, perfect square trinomial, class 9 mathematicsView options
\(x^2+2xy+y^2=(x+y)^2\)
\(x^2-y^2=(x-y)(x+y)\)
\(x^2-2xy+y^2=(x-y)^2\)
\(x^2+y^2=(x+y)^2\)
Easy · Level 73 · rearrangement,area preservation,difference of squaresView options
(a^2+b^2)
(2ab)
(a^2-b^2)
(a+b)
Question 1EasyLevel 73
In a rectangle with sides (x+2) and (x+6), what will be the area of the small corner?
Correct answer: B
When the rectangle is split into parts of lengths \(x\), \(2\), \(x\), and \(6\), the small corner is formed by the numerical sides \(2\) and \(6\). Therefore, its area is \(2\times 6=12\). The terms \(2x\) and \(6x\) represent the areas of other rectangular parts, not the small corner. Exam tip: in a visual model, identify both side lengths of each small region before multiplying.
In a square visual model, one large square is divided into four parts: one square of area \(a^2\), another square of area \(b^2\), and two rectangles each of area \(ab\). Which algebraic identity does this model represent?
Correct answer: A
The side of the large square is \(a+b\), so its total area is \((a+b)^2\). Adding the four parts gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Hence, option A is correct. Option B has \(-2ab\), which occurs in the square of a difference. Exam tip: two \(ab\) rectangles in an area model give the middle term \(+2ab\).
A square model has side (2x+1). What will be its total area?
Correct answer: B
The area of a square equals the square of its side. Thus, the area is \((2x+1)^2\). Using \((a+b)^2=a^2+2ab+b^2\), where \(a=2x\) and \(b=1\), gives \(4x^2+4x+1\). Option C omits the middle term \(2ab=4x\). Exam tip: while squaring a binomial, always include twice the product of the two terms, \(2ab\).
If one side of a rectangle model is (t) and the other is (t+7), what will be the area?
Correct answer: A
The area of a rectangle is the product of its two sides. Hence, the area is \(t(t+7)\). Using the distributive property, \(t(t+7)=t\cdot t+t\cdot7=t^2+7t\). Therefore, option A is correct. In \(t^2+7\), 7 has not been multiplied by \(t\). Exam tip: when multiplying by brackets, multiply the outside term by every term inside the bracket.
A side of a square is divided into two parts, \(a\) and \(b\). Its area model contains \(a^2\), \(b^2\), and two rectangles of area \(ab\). Which identity does this diagram represent?
Correct answer: A
Each side of the large square is \(a+b\), so its total area is \((a+b)^2\). The four parts of the model have areas \(a^2\), \(ab\), \(ab\), and \(b^2\). Their sum is \(a^2+2ab+b^2\), so the identity is \((a+b)^2=a^2+2ab+b^2\). In \((a-b)^2\), the middle term is \(-2ab\), whereas both \(ab\) rectangles are added in this model. Exam tip: combine the two equal \(ab\) rectangles to obtain \(2ab\).
Which algebraic expression is represented by a visual area model that divides a large square into one \(x^2\) square, two \(xy\) rectangles, and one \(y^2\) square?
Correct answer: A
The two \(xy\) rectangles give the middle term \(2xy\), so the total area is \(x^2+2xy+y^2=(x+y)^2\). In \((x-y)^2\), the middle term is \(-2xy\). Exam tip: add the areas of all regions.
A rectangular area model is divided into four parts with areas \(x^2\), \(3x\), \(2x\), and \(6\). Which factorised expression does this model represent?
Correct answer: A
The side lengths are \(x+3\) and \(x+2\), giving \(x^2+3x+2x+6\), exactly the four regions. In \((x+3)^2\), both middle regions would be \(3x\). Exam tip: read the side lengths from adjoining strips.
If the visual expansion of a square is (w^2+6w+9), what will be the outer side?
Correct answer: B
\(w^2+6w+9=w^2+2\times w\times3+3^2=(w+3)^2\). Hence, the outer side of the square is \(w+3\). If the side were \(w+6\), its expansion would be \((w+6)^2=w^2+12w+36\), so it is not correct. Exam tip: for a perfect-square trinomial, use the square roots of the first and last terms and verify the middle term with \(2ab\).
A rectangle has length (2a+1) and breadth (a). What is the expansion of its area model?
Correct answer: A
The area of a rectangle equals length × breadth. Thus, \(a(2a+1)=a\times 2a+a\times 1=2a^2+a\). Therefore, \(2a^2+a\) is correct. In \(2a^2+1\), the term \(a\), obtained by multiplying \(a\) by 1, has been omitted. Exam tip: when expanding brackets, multiply the outside term by every term inside the bracket.
In a difference of squares model, if the big square has side (9) and the small square has side (2), what is the remaining area?
Correct answer: A
The remaining area is obtained by subtracting the small square’s area from the large square’s area: \(9^2-2^2=81-4=77\). Hence, 77 is correct. 81 is only the area of the larger square and does not account for the removed smaller square. Exam tip: for a remaining area, subtract the removed figure’s area from the larger figure’s area.
If an area model has four regions with areas \(x^2\), \(xy\), \(xy\), and \(y^2\), and together they form one large square, which identity does the model represent?
Correct answer: A
The two \(xy\) rectangles combine to give \(2xy\). Thus the total area is \(x^2+2xy+y^2\), the area of a square with side \(x+y\). Exam tip: always check the sign of the middle term.
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