The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.
Answers
SOLUTION :
Let the shorter side of the rectangular field be 'x' meters.
Longer side be (x + 30) m
Length of the diagonal be (x + 60) m
A.T.Q
In rectangle, the diagonal divides the rectangle into two right angled triangles.
By using Pythagoras Theorem,
H² = P² + B²
(Diagonal)² = (Smaller Side)² + (Longer Side)²
(x + 60)² = (x)² + (x + 30)²
[x² + 60² + 2x × 60 ] = x² + [x² + 30² + 2 x × 30 ]
[(a + b) ² = a² + b² + 2ab]
x² + 120x + 3600 = x² + x² + 60x + 900
x² + 120x + 3600 = 2x² + 60x + 900
2x² - x² + 60x - 120x + 900 - 3600 = 0
x² - 60x - 2700 = 0
x² - 90x + 30x - 2700 = 0
[By middle term splitting]
x(x - 90) + 30(x - 90) = 0
(x - 90) (x + 30) = 0
(x - 90) = 0 or (x + 30) = 0
x = 90 or x = - 30
Since, side can't be negative ,so x ≠ - 30 .
Therefore , length of the shorter side is 90 m.
The length of the longer side = (x + 30) = 90 + 30 = 120 m
Hence, the length of the longer side (length) is 120 m & the length of the shorter side(breadth) is 90 m.
HOPE THIS ANSWER WILL HELP YOU….
Answer :
Shorter side = 90 m
Longer side = 120 m
Step-by-step explanation :
Let ABCD be a rectangular field. And AC be its diagonal.
Let the shorter side be x metres.
i.e., AB = x metres.
It is given that ;
The diagonal of a rectangular field is 60 meters more than the shorter side.
AC = AB + 60
AC = x + 60
Also, longer side is 30 meters more than the shorter side.
BC = AB + 30
BC = x + 30
In ∆ABC ,
∠B = 90° [∵ All angles of rectangle are right angles]
Hence, ABC is a right angled triangle.
By Pythagoras theorem ;
Hypotenuse² = Base² + Height²
AC² = AB² + BC²
(x + 60)² = x² + (x + 30)²
x² + 60² + 2 * x * 60 = x² + x² + 30² + 2 * x * 30
x² + 3600 + 120x = x² + x² + 900 + 60x
x² + 3600 + 120x - x² - x² - 900 - 60x = 0
x² - x² - x² + 120x - 60x + 3600 - 900 = 0
- x² + 60x + 2700 = 0
0 = x² - 60x - 2700
⇒ x² - 60x - 2700 = 0
By splitting the middle term ;
x² - 90x + 30x - 2700 = 0
x ( x - 90 ) + 30 ( x - 90 ) = 0
( x - 90 ) ( x + 30 ) = 0
x = 90 or x = - 30
So, x = 90 and x = - 30 are the roots.
Since, x is the length, it can not be negative.
Therefore, x = 90
∴ Shorter side = x = 90 metres.
Longer side = x + 30
= 90 + 30
= 120 metres.
