Question

the diagonal of a rectangular feild abcd is x+9m and the sides are x+7m and xm fid the value​

Answer 1

Question :

The diagonal of a rectangular field abcd is (x+9) m and the sides are (x+7) m and x m find the value of :-

• Dimensions of the Rectangle

• Area of the Rectangle

Given :

• Diagonal of the Rectangle = (x + 9) m

• Length of the Rectangle = (x + 7) m

• Breadth of the Rectangle = x m

To find :

• Dimensions of the Rectangle

• Area of the Rectangle

Solution :

According to the given information , we can use only the formula for Diagonal of a Rectangle to find the value of x.

We know the formula for Diagonal of a Rectangle , i.e,

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

Where :

• D = Diagonal of the Rectangle
• l = Length of the Rectangle
• b = Breadth of the Rectangle

Now using the formula for Diagonal of a Rectangle and substituting the values in it,we get :

$:\implies \bf{D = \sqrt{l^{2} + b^{2}}}$

$:\implies \bf{(x + 9) = \sqrt{(x + 7)^{2} + x^{2}}}$

By squaring on both the sides , we get :

$:\implies \bf{(x + 9)^{2} = \big[\sqrt{(x + 7)^{2} + x^{2}}\big]^{2}}$

$:\implies \bf{(x + 9)^{2} = (x + 7)^{2} + x^{2}}$

Now using the identity :

(a + b)² = a² + 2ab + b² , we get :

$:\implies \bf{x^{2} + 2 \times x \times 9 + 9^{2} = x^{2} + 2 \times x \times 7 + 7^{2} + x^{2}}$

$:\implies \bf{x^{2} + 18x + 81 = x^{2} + 14x + 49 + x^{2}}$

$:\implies \bf{x^{2} + 18x + 81 = 2x^{2} + 14x + 49}$

$:\implies \bf{0 = (2x^{2} + 14x + 49) - (x^{2} + 18x + 81)}$

$:\implies \bf{0 = 2x^{2} + 14x + 49 - x^{2} - 18x - 81}$

$:\implies \bf{0 = x^{2} - 4x - 32}$

By using the middle-splitting theorem, we get :

$:\implies \bf{0 = x^{2} - (8 - 4)x - 32}$

$:\implies \bf{0 = x^{2} - 8x + 4x - 32}$

$:\implies \bf{0 = x(x - 8) + 4(x - 8)}$

$:\implies \bf{0 = (x - 8)(x + 4)}$

$:\implies \bf{0 = (x - 8 = 0) ; (x + 4 = 0)}$

$:\implies \bf{(x = 8) ; (x = - 4)}$

$:\implies \bf{x = 8 ; (-4)}$

$\boxed{\therefore \bf{x = 8 ; (-4)}}$

Hence, the value of x is 8 and (-4).

But since the dimension of a Rectangle can't be negative (here , -4) , the value of x is 8.

To find the Length and Breadth of the Rectangle :

By substituting the value of x in the given values of length and breadth (in terms of x) , we get :

$:\implies \bf{Length\:(l) = (x + 7)\:m}$

$:\implies \bf{l = (8 + 7)}$

$:\implies \bf{l = 15}$

$\boxed{\therefore \bf{Length\:(l) = 15\:m}}$

Hence the length of the Rectangle is 15 m.

$:\implies \bf{Breadth\:(b) = x\:m}$

$:\implies \bf{b = 8}$

$\boxed{\therefore \bf{Breadth\:(b) = 8\:m}}$

Hence the breadth of the Rectangle is 8 m

To find the area of the Rectangle :

Using the formula for area of the Rectangle and substituting the values in it, we get :

$:\implies \bf{A = Length \times Breadth}$

$:\implies \bf{A = 15 \times 8}$

$:\implies \bf{A = 120}$

$\boxed{\therefore \bf{Area\:(A) = 120\:m^{2}}}$

Hence the area of the Rectangle is 120 m².