Question

3. A wire is in the shape of a square of side 8 cm. If the wire is rebent into a rectangle of length
10 cm, find its breadth. Which of the two shapes encloses larger area?​

Answer 1

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$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(4,0){2}{\line(0,1){4}}\multiput(0,0)(0,4){2}{\line(1,0){4}}\put(-0.5,-0.5){\bf D}\put(-0.5,4.2){\bf A}\put(4.2,-0.5){\bf C}\put(4.2,4.2){\bf B}\put(1.5,-0.6){\bf\large 8\ cm}\put(4.4,2){\bf\large 8\ cm}\end{picture}$

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$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(5,0){2}{\line(0,1){3}}\multiput(0,0)(0,3){2}{\line(1,0){5}}\put(0.03,0.02){\framebox(0.25,0.25)}\put(0.03,2.75){\framebox(0.25,0.25)}\put(4.74,2.75){\framebox(0.25,0.25)}\put(4.74,0.02){\framebox(0.25,0.25)}\multiput(2.1,-0.7)(0,4.2){2}{\sf\large 10 cm}\multiput(-1.4,1.4)(6.8,0){2}{\sf\large 6 cm}\put(-0.5,-0.4){\bf A}\put(-0.5,3.2){\bf D}\put(5.3,-0.4){\bf B}\put(5.3,3.2){\bf C}\end{picture}$

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$\large\mathfrak \pink{Solution:-}$

$\underline \mathbb{GIVEN:-}$

Side of the square = 8 cm

Length of the rectangle = 10

$\underline \mathbb{TO \: FIND:-}$

Which of the two shapes encloses larger area?

$\underline \mathbb{FORMULA:-}$

Perimeter of a square = 4 × sides

Perimeter of a rectangle = 2(length + breadth)

Area of a square = $side^{2}$

Area of a rectangle = Length × Breadth

$\underline \mathbb{BY \: THE \: PROBLEM:-}$

Perimeter of the square = Perimeter of the rectangle ( Because the same wire is used in both cases. )

$Perimeter \: of \: a \: square \: = 4 \: × \: sides \\ \\ \implies \: Perimeter \: of \: a \: square \: = 4 \: × 8 \: cm = 32 \: cm$

$Perimeter \: of \: a \: rectangle \: = \: 2(length + breadth) \\ \\ \implies \: 32 = \: 2(10 + breadth)\\ \\ \implies \cancel{\frac{32}{2} } = (10 + breadth) \\ \\ \implies \: 16 = 10 + breadth \\ \\ \implies \: breadth = 16 - 10 = 6 \: cm$

$Area \: of \: the \: square = {(side)}^{2} = {8}^{2} = 64 {cm}^{2}$

$Area \: of \: the \: rectangle \: = (length \times breadth) = (10 \times 6) {cm}^{2} = 60 \: {cm}^{2}$

$\underline \mathbb{ANSWER:-}$

$\boxed{The \:square \:encloses\: larger\: area. }$