6. 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. 7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

6. 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. 7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

 Let shorter side  = x m

Given that diagonal of a rectangular field is 60 metres more than the shorter side

So diagonal will  = x + 60 m

Given that the longer side is 30 metres more than the shorter side

Longer side = x + 30  m

All angles in rectangle is always = 90 so that we can Use Pythagoras theorem

Diagonal ²  =  shorter side ²  +  longer side ²

Plug the values In this formula we get

(x+60)² = (x)²   + (x+30)²

Now use the formula (a+b)² =  a² +b² +2ab

We get

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

subtract x`2  + 120 x  + 3600  both side we get

0 = x²  - 60 x - 2700

Factorize it now

x ² -  90 x + 30 x – 2700  = 0

x ( x - 90 ) +  30 (x – 90 )=0

( x – 90 ) (x + 30 ) = 0

x- 90 = 0 or x + 30 = 0 

x = 90   or x  = - 30 any side of rectangle cannot negative

  

so shorter side  = x + 30 = 90 + 30 = 120

Diagonal is  = x + 60 = 90 + 60 = 150