Question

In the diagram, the area of the smaller square is$10cm^{2}$. Find the area of the larger square.

Answer 1

ANSWER:

• Area of larger square = 20 cm².

GIVEN:

• Area of smaller square = 10 cm².

TO FIND:

• The area of larger square.

EXPLANATION:

From the picture diagonal of the smaller square is equal to the diameter of the circle which is again equal to the side of the larger square.

» Area of smaller square = 10 cm².

» Diagonal length = √2 a (From phythagoras theorem)

★ Area of square = a² sq.units.

» a² = 10

» a = √(10) cm

» Diagonal length = √2 × √(10) = √(20) cm

» Diagonal length = Side of larger square

★ Area of square = a² sq.units.

» Area of larger square = [ √(20) ]²

•°• Area of larger square = 20 cm².

Answer 2

Given :-

In the above figure the area of the smaller square is 10cm² .

To Find :-

The Area of the larger square .

Used Concepts :-

• Area of a square is Side² .
• All sides of square are equal .
• All angles of a square are equal and of 90° .
• The longest chord of a circle is diameter .
• The diagonals of a square are of equal length.
• Pythagoras Theorem .

Solution :-

Lets say the sides of the Smaller square be JM , JK , KL and ML and That if larger square be J'K' , J'M' , M'L' and K'L' respectively .

Then , it is easily understandable that JL and MK are the longest chord of the given circle . Since , the diameter of the circle is JL .

For diagram kindly see the attachment .

Now , At first we need the Side of the smaller square .

=> In square JKLM ,

=> JK² = 10

=> JK = √10 cm .

As we knows that , all angles of a square are of 90° and the side opposite to 90° is the hypotenuse . So In right angled triangle KML ,

=> By Pythagoras Theorem ,

=> KM² = ML² + KL²

=> KM² = ( √10 )² + ( √10 )²

{ Because , All sides of square are equal } .

=> KM² = 10 + 10

=> KM² = 20

=> KM = √20

=> KM = √ 2 × 2 × 5

=> KM = 2√5 cm

As The length of diagonals of square are equal. So , KM = JL

Hence , the diameter of the circle is JL = 2√5 cm .

Now , Let us consider two points on the circle N and X respectively . It is easily seen that , NX is the diameter of the circle and as the circle touches the square at N and X . So , NX is also the side of the square .

As , NX is diameter ,

So , NX = JL = 2√5 cm

And NX is also the side of the square .

So , The Area of larger square = Side² i.e NX² :-

=> NX² = ( 2√5 )²

=> NX² = 20 cm²

Henceforth , Our required answer is 20cm² .