Question

A square and the rectangle have the same perimeter 100cm.find the side of the square if the rectangle has a breadth 2cm less than that of a square.find the length breadth and area of the rectangle

Answer 1

Question :

A square and the rectangle have the same perimeter 100cm. Find the side of the square .

If the rectangle has a breadth 2 cm less than that of the side of the square, then find the length, breadth and area of the rectangle

Given :

• Perimeter of the Rectangle = Perimeter of the square = 100 cm

• Breadth of the Rectangle = Length of square - 2

To Find :

• The side of the square

• Length of the Rectangle

• Breadth of the Rectangle

• Area of the Rectangle

Solution :

⠀⠀⠀⠀To find the side of the square :

By using the formula for perimeter of a square and substituting the values in it , we get :

$\underline{\boxed{\bf{P = 4 \times a}}}$

Where :-

• P = Perimeter of the square
• a = Equal side of the square.

$:\implies \bf{100 = 4 \times a}$

$:\implies \bf{100 = 4 \times a}$

$:\implies \bf{\dfrac{100}{4} = a}$

$:\implies \bf{25 = a}$

$\underline{:\implies \bf{Side\:(a) = 25\:cm}}$

Hence, the side of the square is 25 cm

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⠀⠀⠀To find the breadth of the Rectangle :

According to the Question , the breadth of the Rectangle is 2 less than the side of the square.

So , the equation formed is :-

$\boxed{:\implies \bf{Breadth\:of\:Rectangle = Side\:of\:Square - 2}}$

Now , Substituting the value of side of the square in the equation , we get :

$:\implies \bf{Breadth\:of\:Rectangle = 25 - 2}$

$\underline{\therefore \bf{Breadth\:of\:Rectangle = 23\:cm}}$

Hence, the length of the Rectangle is 23 cm.

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⠀⠀⠀⠀To find the length of the Rectangle :

From the given Perimeter and the known value of the breadth of the Rectangle , and by using the formula for perimeter of a Rectangle , we can find the value of the length of the Rectangle.

Let the length of the Rectangle be l cm.

$\underline{\boxed{\bf{P = 2(l + b)}}}$

Where :-

• P = Perimeter of the Rectangle
• l = Length of the Rectangle
• b = Breadth of the Rectangle

$:\implies \bf{P = 2(l + b)}$

$:\implies \bf{100 = 2(l + 23)}$

$:\implies \bf{100 = 2l + 46}$

$:\implies \bf{100 - 46 = 2l}$

$:\implies \bf{54 = 2l}$

$:\implies \bf{\dfrac{54}{2} = l}$

$:\implies \bf{27 = l}$

$\underline{\therefore \bf{length\:(l) = 27\:cm}}$

Hence, the length of the Rectangle is 26 cm.

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⠀⠀⠀⠀⠀⠀To find the area of the Rectangle :

By using the formula for area of rectangle and Substituting the values in it , we get :

$\underline{\boxed{\bf{A = Length \times Breadth}}}$

$:\implies \bf{A = 27 \times 23}$

$:\implies \bf{A = 621}$

$\underline{\therefore \bf{Area\:(A) = 621\:cm^{2}}}$

Hence, the area of the Rectangle is 621 cm².

Answer 2

$\small \underline{\underline{\sf Given}}\begin{cases} \sf \bull perimeter \: of \: square = perimeter \: of \: rectangle = 100m \\ \sf \bull breadth \: of \: recangle \: is \: 2cm \: less \: than \: side \: of \: square \end{cases}$

$\rule{300}{2}$

$\underline{\underline{\sf Find}}$

• Side of square
• Length and Breadth of Rectangle
• Area of Rectangle

$\rule{300}{2}$

$\underline{\underline{\sf Solution}}$

we, know that

$\underline{\boxed{\sf Perimeter \: of \: Square = 4 \times (side)}}$

where,

• Perimeter of Square = 100cm

So,

$\sf \to Perimeter \: of \: Square = 4 \times (side) \\ \\ \sf : \implies 100 = 4 \times (side) \\ \\ \sf : \implies \dfrac{100}{4} = side \\ \\ : \implies \sf side = 25cm$

Hence, Side of square = 25cm

$\rule{300}{2}$

Now,

In Question it is said that Breadth of Rectangle is 2cm less than Side of Square

So, Breadth = Side of square - 2

where,

• Side of square = 25cm

So,

Breadth = 25 - 2 = 23cm

Hence, Breadth of Rectangle = 23cm

$\rule{300}{2}$

Now, we know that

$\underline{\boxed{\sf Perimeter \: of \: rectangle = 2(l + b)}}$

where,

• Perimeter of Rectangle = 100m
• Breadth of Rectangle = 23cm

So,

$\sf \to Perimeter \: of \: rectangle = 2(l + b) \\ \\ \sf : \implies 100 = 2(l + 23)\\ \\ \sf : \implies 100 = 2l + 46\\ \\ \sf : \implies 100 - 46 = 2l \\ \\ \sf : \implies 54 = 2l\\ \\ \sf : \implies l = \dfrac{54}{2} = 27cm \\ \\ \sf : \implies l = 27cm$

Hence, Length of Rectangle = 27cm

$\rule{300}{2}$

we, know that

$\underline{\boxed{\sf Area \: of \: rectangle = l \times b}}$

where,

• Length = 27cm
• Breadth = 23cm

So,

$\sf \to Area \: of \: rectangle = l \times b\\ \\ \sf : \implies Area \: of \: rectangle = 27 \times 23 \\ \\\sf : \implies Area \: of \: rectangle = 621{cm}^{2}$

Hence, Area of Rectangle = 621cm²

$\rule{300}{2}$

Solution:

• Side of square = 25cm
• Breadth of Rectangle = 23cm
• Length of Rectangle = 27cm