Question

in a rectangle,difference between length and breadth is 7cm.if length of its diagonalis 13cm.then find the area of the rectangle​

Answer 1

Given

Difference b/w length & breadth = 7 cm.

Length of diagonal = 13 cm.

To find

Area of rectangle

Solution

Let the length & breadth be l & b m respectively.

According to 1st case :

⟼ l - b = 7

⟼ l = b + 7 -eq.( l )

We know that,

➥ Diagonal = √(Length² + Breadth²)

Putting values from eq.( l ) :

➻ 13 = √[(b + 7)² + b²]

➻ 13 = √[b² + 14b + 49 + b²]

➻ 13 = √[2b² + 14b + 49]

• Squaring both sides :

➻ 13² = 2b² + 14b + 49

➻ 2b² + 14b + 49 - 169 = 0

➻ 2b² + 14b - 120 = 0

➻ 2(b² + 7b - 60) = 0

➻ b² + 7b - 60 = 0

➻ b² + 12b - 5b - 60 = 0

➻ b(b + 12) - 5(b + 12) = 0

➻ (b - 5)(b + 12) = 0

➻ b = 5 or, b = -12

As, breadth of rectangle can't be negative.

Therefore, breadth of rectangle = 5 cm.

Now putting this value in eq.( l ) :

⟶ l = 5 + 7

⟶ l = 12

Therefore, length of rectangle = 12 cm.

Now finding area of rectangle :

➥ Area of rectangle = l × b

Putting values :

➙ Area of rectangle = 12 × 5

➙ Area of rectangle = 60 cm²

Therefore,

Area of rectangle = 60 cm² .

Answer 2

Given,

• In a Rectangle , Difference between Length and Breadth is 7 cm.
• If the length of its Diagonal is 13 cm.

To Find,

• The Area of Rectangle .

Solution :

⇒Suppose the Length be "a"

And, Suppose the breadth be "b"

$\mapsto \underline{\sf{\pink{According \ to \ the \ First \ Condition :}}}$

• The Difference Between Length and Breadth is 7 cm

$\implies \sf{a - b = 7}$

$\implies \boxed{\sf{a = 7 + b }}$

$\mapsto \underline{\sf{\pink{According \ to \ the \ Second \ Condition :}}}$

• The Length of its Diagonal is 13 cm

From Pythagoras Theorem :-

$\implies \boxed{\sf{\red{(Hypotenuse)^{2} = (Perpendicular)^{2} + (Base)^{2}}}}$

$\implies \sf{(13)^{2} = a^{2} + b^{2}}$

$\implies \sf{169 = a^{2} + b^{2}}$

$\implies \sf{(7 + b)^{2} + b^{2} = 169}$

$\implies \sf{49 + b^{2} + 14b + b^{2} = 169}$

$\implies \sf{2b^{2} + 14b = 169 - 49}$

$\implies \sf{2b^{2} + 14b - 120 = 0}$

$\implies \sf{2(b^{2} + 7b - 60) = 0}$

$\implies \sf{b^{2} + 7b - 60 = 0}$

$\implies \sf{b^{2} + 12b - 5b - 60 = 0}$

$\implies \sf{b(b + 12) -5(b + 12) = 0}$

$\implies \sf{(b - 5) (b + 12) = 0}$

$\implies \sf{\therefore b = 5 \ or \ b = - 12}$

Therefore , Side Cannot be negative so b = 5

Now Put the Value of b in First Condition :-

$\implies \sf{a = 7 + b}$

$\implies \sf{a = 7 + 5}$

$\implies \boxed{\sf{a = 12}}$

Now We have Length And Breadth So Find Area of Rectangle :-

$\implies \sf{Area \ of \ Rectangle = Length \ \times \ Breadth}$

$\implies \sf{Area \ of \ Rectangle = 12 \times 5}$

$\implies \boxed{\sf{\pink{Area \ of \ Rectangle = 60 \ cm^{2}}}}$

$\rule{200}2$