Question

In the given figure PQRS is a rectangle inscribed in the circle. If PS = 5 cm and PQ = 12 cm, find the
area of un-shaded region. (Use π =
22
/7
)​

Answer 1

Extra Constructions:

A diagonal QS connecting sides Q and S. [Please refer the attachment to view the diagram]

Solution:

To find the unshaded portion, we need to find the measure of the diagonal of the rectangle as:

Diagonal = Diameter of the circle

Further calculations will help us find the area.

Solving:

As we have the diagonal, consider ΔQRS.

QR = 5 cm

RS = 12 cm

QS = ?

According to the Pythagoras theorem:

Altitude² + Base² = Hypotenuse²

5² + 12² = Hypotenuse²

25 + 144 = Hypotenuse²

Hypotenuse² = 169

Hypotenuse = √169 = 13 cm

As Diagonal = Diameter,

Diameter = 13 cm

Radius = 13/2 cm

Now, we need to find the area of the rectangle and the area of the circle.

Area of the rectangle

Length = 12 cm

Breadth = 5 cm

Area = 12 x 5 = 60 cm²

Area of the complete circle

Radius = 13/2 cm

Area :

$\sf{=\dfrac{22}{7}\times\dfrac{13}{2}\times{13}{2}}$

$\sf{=\dfrac{1859}{14}\:cm^2}$

$\sf{=132.78\:cm^2}$

Area of the unshaded portion:

Area of the circle - Area of the rectangle

$\sf{=132.78-60}$

$\sf{=72.78\:cm^2}$

∴ Area of the unshaded portion is 72.78 cm².

Answer 2

Answer:

Correct option is

A

14

1019π

cm

2

Take △PQS

Applying pythaooras theorem, we get

PQ

2

+PS

2

=QS

2

⇒12

2

+5

2

=QS

2

⇒QS

2

=169

⇒QS=13

QS is a diameter to the circle.

Hence, radius of a circle will be

2

13

cm

Area of rectangle =l×b

=5×12=60 sq. cm

Area of circle =πr

2

=

7

22

×(

2

13

)

2

=

14

1859

sq. cm

Therefore, area of the shaded region =Area of circle−Area of rectangle=

14

1859

−60=

14

1019

sq. cm