Name: Area of Annular Regions
Uploaded: 2019-07-19
Duration: 6 min 24 s
Description: The annular region of two given circles is the area enclosed between the concentric circles. In this video, we will discuss this with the help of a few examp...

Question

In the adjoining figure, $A B C D$ is a square drawn inside a circle with centre $O$ . The centre of the square coincides with $O$ and the diagonal $A C$ is horizontal. If $A P, D Q$ are vertical and $A P = 45 cm, D Q = 25 cm$ , find the side of the square.

Accepted Answer

Given, $A B C D$ is a square drawn inside a circle with centre $O$ . The centre of the square coincides with $O$ and the diagonal $A C$ is horizontal

$A P, D Q$ are vertical and $A P = 45 cm, D Q = 25 cm$ .

We know that diagonals of a square bisect each other at right angle.

$∴ A O = O C = O B = O D$ _________ $(1)$

Let the radius of the circle be $r cm$ .

Radius $= r = O Q = O D + D Q = (O B + 25) cm = (O A + 25) cm$ [From $(1)$ ]

Join $P O$ .

Here, $P O$ is also a radius of the circle.

$∴ P O = (A O + 25) cm$

Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In right-angled $A P, D Q$ ,

$O P^{2} = A P^{2} + A O^{2}$

$\Rightarrow {(A O + 25)}^{2} = 45^{2} + O A^{2}$

$\Rightarrow A O^{2} + 50 A O + 625 = 2025 + O A^{2}$

$\Rightarrow 50 A O = 2025 - 625$

$\Rightarrow 50 A O = 1400$

$\Rightarrow A O = \frac{1400}{50}$

$\Rightarrow A O = 28$

In right-angled $△ A O B$ ,

$A B^{2} = O A^{2} + O B^{2}$

$\Rightarrow A B^{2} = O A^{2} + O A^{2}$ [From $(1)$ ]

$\Rightarrow A B^{2} = 2 O A^{2}$

$\Rightarrow A B^{2} = 2 {(28)}^{2}$

$\Rightarrow A B^{2} = 2 \times 784$

$\Rightarrow A B = \sqrt{1568}$

$\Rightarrow A B = 39.58$

Therefore, the side of the square is $39.58 cm$ .

In the adjoining figure, $A B C D$ is a square drawn inside a circle with centre $O$ . The centre of the square coincides with $O$ and the diagonal $A C$ is horizontal. If $A P, D Q$ are vertical and $A P = 45 cm, D Q = 25 cm$ , find the side of the square.

Important Questions on Areas Related to Circles

Learn and Prepare for any exam you want !

In the adjoining figure, ABCD is a square drawn inside a circle with centre O. The centre of the square coincides with O and the diagonal AC is horizontal. If AP, DQ are vertical and AP=45 cm, DQ=25 cm, find the side of the square.

Important Questions on Areas Related to Circles

Learn and Prepare for any exam you want !

In the adjoining figure, $A B C D$ is a square drawn inside a circle with centre $O$ . The centre of the square coincides with $O$ and the diagonal $A C$ is horizontal. If $A P, D Q$ are vertical and $A P = 45 cm, D Q = 25 cm$ , find the side of the square.