Question

The diagonal of a rectangular ground is 60 meters more than the breadth of the ground. If the length of the ground is 30 meters more than the breadth, find the area of the ground.

Answer 1

Dear student,

Answer:Area of rectangular park ) = 10,800 $m^{2}$

Solution:

Let the breadth of the rectangular ground is x meters

than diagonal of rectangular ground will be 60+x meters

now length of rectangular park is x+30 meters

Area of rectangle = length x breadth

= x( x+30) sq-meter

In terms of diagonal ; Area of rectangle = $= l \sqrt{d^{2} - l^{2}} \\</strong> \\ \\ =( x+30)\sqrt{(x+60)^{2}- (x +30)^{2}} \\ \\ =(x+30)\sqrt{x^{2}+120x+3600-x^{2}-60x-900} \\ \\ =(x+30)\sqrt{60x+2700}$

Since area is equal ,so

x(x+30) = (x+30) ($\sqrt{60x+2700}$

Taking square both sides

$x^{2} = 60x+2700\\ \\ x^{2} -60x-2700 =0$

Do factorisation for finding the roots of quadratic equation

$x^{2} - 90x +30x -2700 =0\\ \\ x(x -90) + 30(x - 90) =0\\ \\ (x-90)(x+30) =0\\ \\ x-90 =0 \\\\ x =90\\ \\ x+30 = 0 \\ \\ x = -30$

Discard x = -30, since breadth cannot be negative quantity.

So, Breadth of rectangular park is 90 meters

So, length is x+ 30 = 90+ 30 = 120 meters

Area will be = length x breadth

Area = 90 (120) = 10,800 $m^{2}$

Hope it helps you.

Answer 2

HELLO DEAR,


Let the shorter side of the rectangular field be x meters.
so,the longer side will be(x + 30) meters.

And the length of the diagonal will be (x + 60) meters.

Now, according to the question

The diagonal divides the rectangular into two right angled triangles and the diagonal is the common side of the two triangles and it is also the longest side of the triangles i.e. the hypotenuse.

So, by Pythagoras Theorem,
(Diagonal)² = (Smaller Side)² + (Longer Side)²

i.e. (AC)² = (AB)² + (BC)²

(x + 60)² = (x)² + (x + 30)²

x² + 120x + 3600 = x² + x² + 60x + 900

x² + 60x - 120x + 900 - 3600 = 0

x² - 60x - 2700 = 0

x² - 90x + 30x - 2700 = 0

x(x - 90) + 30(x - 90) = 0

(x - 90) (x + 30) = 0

x = 90 and x = - 30
x = -30 [ as negative length cannot be possible ].

So,
the length of the shorter side is 90 meters. and the length of the longer side is 90 + 30 = 120 meters.
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I HOPE ITS HELP YOU DEAR,
THANKS