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The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field. - Mathematics

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The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

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Solution 1

Let the shorter side of the rectangle be x m.

Then, larger side of the rectangle = (x + 30) m

Diagonal of rectangle = `sqrt(x^2+(x+30)^2)`

It is given that the diagonal of the rectangle = (x+30)m

`:.sqrt(x^2+(x+30)^2) = x +60`

⇒ x2 + (x + 30)2 = (x + 60)2

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

⇒ x2 - 60x - 2700 = 0

⇒ x2 - 90x + 30x - 2700 = 0

⇒ x(x - 90) + 30(x -90)

⇒ (x - 90)(x + 30) = 0

⇒ x = 90, -30

However, side cannot be negative. Therefore, the length of the shorter side will be 90 m.

Hence, length of the larger side will be (90 + 30) m = 120 m.

Solution 2

Let the length of smaller side of rectangle be x meters then larger side be (x + 30) meters and their diagonal be (x + 60)meters

Then, as we know that Pythagoras theorem

x2 + (x + 30)2 = (x + 60)2

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

2x2 + 60x + 900 - x2 - 120x - 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 = 0

x = 90

Or

x + 30 = 0

x = -30

But, the side of rectangle can never be negative.

Therefore, when x = 90 then

x + 30 = 90 + 30 = 120

Hence, length of smaller side of rectangle be 90 meters and larger side be 120 meters.

Concept: Solutions of Quadratic Equations by Completing the Square
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APPEARS IN

RD Sharma Class 10 Maths
Chapter 4 Quadratic Equations
Exercise 4.1 | Q 2 | Page 64
NCERT Class 10 Maths
Chapter 4 Quadratic Equations
Exercise 4.3 | Q 6 | Page 88
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