Question

the diagonal of rectangular field is 60 M more than the shorter side if the longer side is 30 M more than the shorter side find the side of the field​

Answer 1

✬ Shorter side = 90 m ✬

✬ Longer side = 120 m ✬

Step-by-step explanation:

Given:

• Diagonal of rectangular field is 60 m more than shorter side.
• Longer side of field is 30 m more than shorter side.

To Find:

• What is the meaure of sides of rectangular field ?

Solution: Let ABCD be a rectangular field and BC be the shorter side of x m. Therefore,

➼ AB = Longer side = 30 more than BC

• AB = ( 30 + x ) m

➼ AC = Diagonal = 60 m more than BC

• AC = ( 60 + x ) m

Now, In ∆ABC right angled at B, we have

• BC = x m [ Perpendicular ]
• AB = (30 + x) m [ Base ]
• AC = (60 + x ) m [ Hypotenuse ]

Applying Pythagoras Theorem in ∆ABC:-

★Pythagoras Theorem → H² = B² + P² ★

$\implies{\rm }$ AC² = AB² + BC²

$\implies{\rm }$ (60 + x)² = (30 + x)² + (x)²

By using (a+b)² = a² + b² + 2ab

➮ 60² + x² + 2•60•x = 30² + x² + 2•30•x + x²

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

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

➮ 2700 + 60x = x²

➮ 2700 + 60x –x² = 0

➮ –x² + 60x + 2700 = 0

Factorise it by using middle term splitting method

➟ –x² + 60x + 2700

➟ –x² + 90x – 30x + 2700

➟ –x (x – 90) –30 (x – 90)

➟ (–x –30) (x – 90)

➟ (–x –30) = 0 (x – 90) = 0

➟ x = –30 or x = 90

Since, x is length therefore it cannot be negative. So x = 90 m

∴ Shorter side = BC = x = 90 m

________________________

➬ Longer side = AB = (30 + x)

➬ (30 + 90) = 120m

________________________

➱ Diagonal = AC = (60 + x)

➱ (60 + 90) = 150 m

Answer 2

Let the shorter side be " x "

A/c " the longer side is 30 M more than the shorter side "

⇒ Longer side is " x + 30 "

A/c " the diagonal of rectangular field is 60 M more than the

shorter side  "

⇒ Diagonal =  " x + 60 "

Now apply pythagoras theorem ,

⇒ x² + (x+30)² = (x+60)²

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

⇒ x² - 60x - 2700 = 0

⇒ x² + 30x - 90x  - 2700 = 0

⇒ x ( x + 30 ) - 90 ( x + 30 ) = 0

⇒ ( x - 90 ) ( x + 30 ) = 0

⇒ x = 90 & x = -30

But lengths can't be negative .

⇒ x = 90

So the shorter side = x = 90 m

Longer side = x + 30 = 120 m