Question

The area of a rectangle reduces by 20 metre square, if its in length is increased by 1 metre and the breadth is reduced by 2 metre . The area increased by 12 metere , Square if the length is reduced by 3 metre and the breadth is increased by 4 metre .Find the dimension of the rectangle.​

Answer 1

Answer:

The length is 15m and Breadth is 12m.

Step-by-step explanation:

Solution :

Let the -

• Length be a
• Breadth be b

Area of the rectangle = Length × Breadth

$\textsf{\underline{\underline{Condition 1 -}}}$

The area of a rectangle reduces by 20m², when its length is increased by 1 metre and the breadth is reduced by 2 metres.

Area of this rectangle =

$\sf{\longrightarrow} \: a \times b = ab \: {m}^{2}$

★ $\boxed{\sf{(a+1) \times (b-2)=(ab - 20)}}$

$\sf{\longrightarrow} \: (a+ 1) \times (b - 2) = (ab - 20) \\ \\ \sf{\longrightarrow} \: \cancel{ab} - 2a + b- 2 = \cancel{ab}- 20 \\ \\ \sf{\longrightarrow} \: - 2a + b - 2 = - 20 \\ \\ \sf{\longrightarrow} \: b= 2a- 20 + 2 \\ \\ \sf{\longrightarrow} \: b = 2a - 18 \: ---- \: \rm{\gray{(1)}}$

$\rule{300}{1.5}$

$\textsf{\underline{\underline{Condition 2 - }}}$

The area increased by 12 m², when the length is reduced by 3m and the breadth is increased by 4m.

★ $\boxed{\sf{(a - 3) \times (b+ 4) = (ab + 12)}}$

$\sf{\longrightarrow} \: (a - 3) \times (b + 4) = (ab + 12) \\ \\ \sf{\longrightarrow} \: \cancel{ab} + 4a - 3b - 12 = \cancel{ab} + 12 \\ \\ \sf{\longrightarrow} \: 4a - 3b = 12 + 12 \\ \\ \sf{\longrightarrow} \: 4a - 3b = 24 \: ---- \: \rm{\gray{(2)}}$

$\rule{300}{1.5}$

Substitute the value of b in Equation 2 -

$\sf{\longrightarrow} \: 4a - 3(2a - 18) = 24 \\ \\ \sf{\longrightarrow} \: 4a - 6a + 54 = 24 \\ \\ \sf{\longrightarrow} \: - 2a = 24 - 54 \\ \\ \sf{\longrightarrow} \: - 2a = - 30 \\ \\ \sf{\longrightarrow} \: a = \dfrac{30}{2} \\ \\ \sf{\longrightarrow} \: a = 15$

Length = 15 m

$\rule{300}{1.5}$

Substitute the value of 'a' in Equation 1 -

$\sf{\longrightarrow} \: b = 2a - 18 \\ \\ \sf{\longrightarrow} \: b = 2(15) - 18 \\ \\ \sf{\longrightarrow} \: b = 30 - 18 \\ \\ \sf{\longrightarrow} \: b = 12$

Breadth = 12 m

$\therefore$ The length is 15m and Breadth is 12m.

Answer 2

||✪✪ QUESTION ✪✪||

The area of a rectangle reduces by 20 metre square, if its in length is increased by 1 metre and the breadth is reduced by 2 metre . The area increased by 12 metere , Square if the length is reduced by 3 metre and the breadth is increased by 4 metre .Find the dimension of the rectangle.

|| ✰✰ ANSWER ✰✰ ||

Let us assume that, initial Length and breadth of Rectangle are x m. and y m . Respectively.

➪ Original Area of Rectangle = Length * Breadth = (x * y)m²

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☙ Case ❶ :-

The area of a rectangle reduces by 20 metre square, if its in length is increased by 1 metre and the breadth is reduced by 2 metre .

➺ New Length = (x + 1) m

➺ New Breadth = (y -2) m

➺ Decreased Area = (xy - 20)m²

So,

➪ (x +1)(y-2) = (xy - 20)

➪ xy -2x + y -2 = xy - 20

➪ (-2x + y) = -20 + 2

➪ -1( 2x - y) = (-1) * 18

➪ 2x - y = 18 -------------------- Equation (1)

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☙ Case ❷ :-

The area increased by 12 metere , Square if the length is reduced by 3 metre and the breadth is increased by 4 metre .

➻ New Length = (x -3)

➻ New Breadth = (y +4)

➻ Increased Area = (xy + 12) m²

So,

➪ (x-3)(y+4) = (xy + 12)

➪ xy +4x -3y -12 = xy + 12

➪ 4x - 3y = 12 + 12

➪ 4x - 3y = 24 -------------------------- Equation(2)

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Now, Multiplying Equation (1) by 2 , and than Subtracting Equation (2) from it , we get,

☛ 2(2x - y) - (4x -3y) = 2*18 - 24

☛ 4x - 2y - 4x + 3y = 36 - 24

☛ y = 12 m.

Putting This value in Equation (1) now,

☞ 2x - 12 = 18

☞ 2x = 18 + 12

☞ 2x = 30

Dividing both sides by 15,

☞ x = 15m.

Hence, initial Length and breadth of Rectangle are 15m and 12m Respectively..

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★★Extra Brainly Knowledge★★

✯✯ Some Properties of Rectangle ✯✯

1) Each of the interior angles of a rectangle is 90°.

2) The diagonals of a rectangle bisect each other.

3) The opposite sides of a rectangle are parallel.

4) The opposite sides of a rectangle are equal.

5) A rectangle whose side lengths are a and b has area = a×b×sin90° = a×b

6) A rectangle whose side lengths are a a and b b has perimeter 2(a + b)...

7) The length of each diagonal of a rectangle whose side lengths are a and b is √(a²+b²)..

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