Question

24) A pair of adjacent sides of a rectangle are in the ratio 3:4. If its diagonal
is 20 cm find the lengths of the sides and hence the perimeter of the
rectangle.​

Answer 1

GiveN:-

A pair of adjacent sides of a rectangle are in the ratio 3:4. If its diagonal is 20 cm.

To FinD:-

Find the lengths of the sides and hence the perimeter of the rectangle.

SolutioN:-

Diagram :

$\setlength{\unitlength}{1cm}\begin{picture}(0,0)\thicklines\multiput(0,0)(5,0){2}{\line(0,1){3}}\multiput(0,0)(0,3){2}{\line(1,0){5}}\put(0.03,2.74){\framebox(0.25,0.25)}\put(4.74,0.01){\framebox(0.25,0.25)}\put(2,-0.7){\sf\large 3x cm}\put(-1.4,1.4){\sf\large 4x cm}\put(-0.5,-0.4){\bf A}\put(-0.5,3.2){\bf D}\put(5.3,-0.4){\bf B}\put(5.3,3.2){\bf C}\qbezier(0,0)(0,0)(5,3)\put(1.65,1.8){\sf\large 20 cm}\end{picture}$

• Let the length be 3x.
• Breadth = 4x

We know that,

$\small{\green{\underline{\boxed{\bf{Diagonal=\sqrt{(Length)^2+(Breadth)^2}}}}}}$

where,

• Diagonal = 20 cm
• Length = 3x
• Breadth = 4x

Putting the values,

$\small\implies{\sf{20=\sqrt{(3x)^2+(4x)^2}}}$

Squaring both the sides,

$\small\implies{\sf{(20)^2=(3x)^2+(4x)^2}}$

$\small\implies{\sf{400=9x^2+16x^2}}$

$\small\implies{\sf{400=25x^2}}$

$\small\implies{\sf{\dfrac{400}{25}=x^2}}$

$\small\implies{\sf{16=x^2}}$

Square rooting both the sides,

$\small\implies{\sf{\sqrt{16}=x}}$

$\large\therefore\boxed{\bf{x=4.}}$

Dimensions:-

1. Length = 3x = 3 × 4 = 12 cm
2. Breadth = 4 × 4 = 16 cm

Now the perimeter:-

$\small{\green{\underline{\boxed{\bf{Perimeter=2(Length+Breadth)}}}}}$

where,

• Length = 12 cm
• Breadth = 16 cm

Putting the values,

$\small\implies{\sf{Perimeter=2(12+16)}}$

$\small\implies{\sf{Perimeter=2\times28}}$

$\large\therefore\boxed{\bf{Perimeter=56\:cm.}}$

The length is 12 cm.

The breadth is 16 cm.

The perimeter is 56 cm.

Answer 2

$\huge \sf \underline \red{Answer : }$

$\sf \underline \purple{ \therefore \: so \: perimeter \:is \: 56cm}$

$\sf \underline \purple{ \therefore \: length = 12cm \: and \: breadth = 16cm }$

$\huge \sf \underline \pink{Given : }$

• The pair of a adjacent sides of a rectangle are in the ratio 3:4

• Diagonal = 20

$\huge \sf \underline \blue{To \: find : }$

• length of the sides

• perimeter of rectangle

$\huge \sf \underline \orange{solution : }$

$\sf \underline{Given \: length : breadth = 3 : 4}$

$\sf \underline{so}$

$\sf\underline{Let \: length \: be \: 3x}$

$\sf \underline{Let \: breadth \: be \: 4x}$

$\sf \underline{diagonal = 20}$

• The Diagonal of two adjacent sides form a right angled triangle by Pythagoras theorem

$\sf \underline{we \: know \: that \: pythagoras \: formula : }$

$\: \: \: \: \: \: \: \sf{ \implies \: (20) {}^{2} = (4x) {}^{2} + (3x) {}^{2}}$

$\: \: \: \: \: \: \: \sf{ \implies \: (400) = (16x) {}^{2} + (8x) {}^{2} }$

$\: \: \: \: \: \: \: \sf{ \implies \: (25) {}^{2} =400 }$

$\: \: \: \: \: \: \: \sf{ \implies \: {x }^{2} = \dfrac{400}{25} }$

$\: \: \: \: \: \: \: \sf{ \implies \: {x }^{2} = 16 }$

$\: \: \: \: \: \: \: \sf{ \implies \: {x } = 4}$

$\: \: \sf \underline{then}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ \star \: length = 4 \times 4 = 16cm}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ \star \: breadth = 3 \times 4 = 12cm}$

$\: \: \: \: \: \sf \underline{Now \: we \: know \: that \: perimeter \: formula}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \bf{ \boxed{ \underline{ \underline{ \red{ \tt{perimeter = 2(l + b) \: }}}}}}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ \implies2(16 + 12)}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ \implies2(28)}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \sf{ \implies56}$

$\sf \underline{ \therefore \: so \: perimeter \:is \: 56cm}$

____________________________________

$\: \: \: \: \: \: \: \: \: \: \: \sf{ \star \: length \: of \: rectangle = 12cm}$

$\: \: \: \: \: \: \: \: \: \: \: \sf{ \star \: breadth \: of \: rectangle = 16cm}$

$\: \: \: \: \: \: \: \: \: \: \: \sf{ \star \: perimeter \: of \: rectangle = 56cm}$