the diagonal of a rectangular field is 30 metres more than the shorter side if the longer side is 15 more than the shorter side find the sides of the field
Answers
ANSWER:-
Given:
The diagonal of a rectangle field is 30m more than the shorter side if the longer side is 15m more than the shorter side.
To find:
The sides of the field.
Solution:
Let ABCD be the rectangle field.
⚫AC is a diagonal.
⚫Let the shorter side, AB= x m
⚫Diagonal is 30m more than shorter side.
So,
=) AC= AB+ 30
=) AC = x +30m
&
Longer side is 15m more than shorter side.
=) BC= AB+ 15
=) BC= x+ 15m
In right angled ∆ABC,
angle B= 90°
Using Pythagoras Theorem:
(Hypotenuse)²=(base)²+(perpendicular)²
=) AC² = AB² + BC²
=) (x+30)²= x² + (x+15)²
=) x²+30²+2× x×30=x²+ x²+15²+ 2× x ×15
=) x²+ 900+ 60x= x² + x²+ 225+ 30x
=) x² +900+60x- x²-x² -225-30x =0
=) x²-x²-x² + 60x-30x+900-225=0
=) -x² +30x + 675=0
=) 0= x² - 30x -675
=) x² -30x -675=0
Now,
We factorise using splitting the middle method term;
=) x² -30x -675=0
=) x² +15x - 45x-675=0
=) x(x+15)- 45(x+15)=0
=) (x+15)(x-45)=0
=) x+15= 0 or x-45=0
=) x= -15 or x= 45
So,
x= -15 & x= 45 are the roots of the equation.
Since x is the length.
It can't be negative.
So,
x= 45.
Therefore,
⚫Shorter side of the field is x= 45m.
&
⚫Longer side of the field is, x+15m
=) 45m + 15m
=) 60m
Hope it helps ☺️
Let the shorter side of the rectangular field be 'x' meters.
Therefore 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
(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 because x = - 30 as 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.