Directions: Study the following questions carefully and choose the right answer.
Important for :
1
If the circumference of a circle is equal to the perimeter of square, then which one of the following is correct ?
» Explain it
C
Let, the radius of a circle = r   and   the side of a square = a

Given, circumference of a circle = perimeter of a square

⇒  2πr = 4a

 ⇒  a = π r = 3.14 r = 1.57r 2 2

Now, Area of the circle = πr2 = 3.14r2

And, Area of the square = a2 = (1.57r2) = 2.4649r2

∴  Area of circle > Area of square

Hence, option C is correct.

2
If the circumference of two circles are in the ratio 2 : 3, then what is the ratio of their areas ?
» Explain it
B
Let the radii of two circles are r1 and r2, respectively.

Given, Circumference of first circle : Circumference of second circle = 2 : 3

⇒  2πr1 : 2πr2 = 2 :3

 ⇒ r1 = 2 r2 3

∴  Area of first circle : Area of second circle = πr12 : πr22

 = ( r1 ) 2 = ( 2 ) 2 = 4 : 9 r2 3

Hence, option B is correct.

3
If the area of a circle inscribed in an equilateral triangle is 154 sq cm, then what is the perimeter of the triangle ?
» Explain it
B
Note: Radius of incircle of an equilateral triangle of side
a =  a .
 2 3

∴  r =  a .
 2 3

Given, area of a circle inscribed in an equilateral triangle = 154 cm2

∴  πr2 = 154 cm2

⇒  π ( a ) 2  = 154
 2 3

⇒   ( a ) 2  =  154  =  154 × 7  = (7)2
 2 3
π 22

⇒ a = 14√3 cm

∴  Perimeter of an equilateral triangle = 3a = 3 × 14√3 = 42√3 cm

Hence, option B is correct.

4
 If the area of a circle is equal to the area of square with side 2 π units, then what is the diameter of the circle ?

» Explain it
C
 Given, side of square (a) = 2 π unit

And, Area of the circle = Area of the square

 ∴  πr2 = a2 = (2 π )2 = 4π

⇒  r2 = 4

⇒  r = 2 units

∴  Diameter of the circle = 2r = 2 × 2 = 4 units

Hence, option C is correct.

5
If the area of a rectangle whose length is 5 units more than twice its width is 75 sq units. What is the perimeter of the rectangle?
» Explain it
A
Let the width of the rectangle (b) = x unit

∴  Length (l) = (2x + 5) units

According to the question,

Area of the rectangle = l × b

∴  75 = (2x + 5) × x

⇒  75 = 2x2 + 5x

⇒  2x2 + 5x – 75 = 0

⇒  2x2 + 15x – 10x – 75 = 0

⇒  x(2x + 15) – 5(2x + 15) = 0

⇒  (x – 5)(2x + 15) = 0

 ⇒  x = 5  and –15 2

Since, width cannot be negative.

∴  width = x = 5 units     and     length = (2x + 5) = 2 × 5 + 5 = 15 units

∴  Perimeter of the rectangle = 2(l + b) = 2 × (15 + 5) = 40 units

Hence, option A is correct.