Question

The perimeter of a rectangle is 42 meters and its diagonals is 15 meters. What are the length of its sides?​

Answer 1

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The perimeter of a rectangle is 42 meters and its diagonals is 15 meters. What are the length of its sides?

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$\bold{\boxed{Perimeter \:of \:rectangle\:=2(length+breadth)}}$

formula for diagonal:-

$\bold{\boxed{D = \sqrt{ {b}^{2} + {l}^{2} } }}$

=>Diagonal is given which is 15metres

There are two solutions for finding length:-

$L1 = \frac{p}{4} + \frac{1}{4} \sqrt{8 {d}^{2} - {p}^{2} }$

$= > \frac{42}{4} + \frac{1}{4} \times \sqrt{ {8 \times (15)}^{2} - {(42)}^{2} }$

$\bold{\red{= > 12m}}$

$L2 = \frac{p}{4} - \frac{1}{4} \sqrt{8 {d}^{2} - {p}^{2} }$

$= > \frac{42}{4} - \frac{1}{4} \sqrt{8 \times {(15)}^{2} - {(42)}^{2} }$

$\bold{\red{= >9m}}$

hence,the length of the sides is either 9m or 12m

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Answer 2

Given :

• Perimeter of rectangle = 42 m
• Diagonal of rectangle = 15 m

To Find :

• Length = ?
• Breadth = ?

Answer :

➻ Perimeter of reactangle = 2(Length + Breadth)

• Let Length be 'l' and Breadth be 'b'.

➻ 42 = 2(l + b)

➻ 42 ÷ 2 = (l + b)

➻ 21 = (l + b) ........[Equation (i)]

Now,by using Phythagoras theorem we get :

➻ (l)² + (b)² = (15)²

➻ l² + b² = 225 ......[Equation (ii)]

Now, Taking equation (i) :

➻ l + b = 21

Squaring both the sides

➻ (l + b)² = (21)²

➻ l² + b² + 2lb = 441

Replacing l² + b² by 225 from equation (ii) :

➻ 225 + 2lb = 441

➻ 2lb = 441 - 225

➻ 2lb = 216

Dividing both sides by 2 we get :

➻ lb = 108 .......[Equation (iii)

Taking equation (i) again we get :

➻ l + b = 21

➻ l = 21 - b ......[Equation (iv)]

Substituting the value of 'l' from equation (iv) in equation (iii) we get :.

➻ (21 - b) (b) = 108

➻ -b² + 21b = 108

➻ -b² + 21b - 108 = 0

By splitting the middle term we get :

➻ -b² -12b - 9b - 108 = 0

➻ -b(b - 12) + 9(b - 12) = 0

➻ (-b + 9) (b -12)

➻ b = 12 m or b = 9 m

• When b = 12 m then Length will be :

➻ l + b = 21 [from equation (i)]

➻ l + 12 = 21

➻ l = 9 m

• When b = 9 m then Length will be :

➻ l + b = 21 [from equation (i)]

➻ l + 9 = 21

➻ l = 12 m