Question

1.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.
2.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.​

Answer 1

1) Let the shorter side be x m .

Then diagonal will be x+60 m .

The longer side will be x+30 m.

By Pythagoras theorem,

(x+60) ²=x ²+(x+30) ²

On simplifying above equation, we get

x ²−60x−2700 = (x−90)(x+30)

x=90,−30

Neglecting negative value, we get

The shorter side is 90 m and the longer side is 120 m .

2) We have,

Let the large number is =x

Square of smaller number is =8x

Now, According to given question,

x² −8x=180

⇒x² −8x−180=0

⇒x² −(18−10)x−180=0

⇒x² −18x+10x−180=0

⇒x(x−18)+10(x−18)=0

⇒(x−18)(x+10)=0

⇒x−18=0,x+10=0

⇒x=18,x=−10

⇒x=−10,18

Either

x=−10 and x=18

x=18 is true (Positive value)

Now, square of small number

=18x=18×8=144

√144

=12

Hence, 12 is the answer.