In the following figure, ABCD is a square of side 2a, Find the ratio between
(i)the circumferences
(ii)the areas of the in circle and the circum-circle of the square.
Answers
Answer:
The ratio of circumferences of inner circle (A1) and circumcircle (A2) is 1 : √2
& Ratio of Areas of inner circle (A1) and circumcircle (A2) is 1 : 2
Step-by-step explanation:
Given :
Side of a square ABCD = 2a
Diameter of incircle = side of a square = 2a
Radius of a incircle ,r = Diameter of incircle/2
r = 2a/2
r = a
Radius of a incircle ,r = a
Diameter of circumcircle = diagonal of a square = √2 side
Radius of circumcircle ,R = √2 side/2
R = (√2 × 2a) /2
R = √2a
Radius of circumcircle ,R = √2a
(i) Ratio of circumferences of inner circle (C1) and circumcircle (C2)
C1 : C2 = 2πr : 2πR
C1 / C2 = 2πr / 2πR
C1 / C2 = r / R
C1 / C2 = a/√2a
C1 / C2 = 1/√2
C1 : C2 = 1 : √2
Ratio of circumferences of inner circle (C1) and circumcircle (C2) = 1 : √2
(ii) Ratio of Areas of inner circle (A1) and circumcircle (A2) :
A1 : A2 = πr² : πR²
A1 / A2 = πr² / πR²
A1 / A2 = r² /R²
A1 / A2 = a²/(√2a)²
A1 / A2 = a²/2a²
A1 / A2 = ½
A1 : A2 = 1 : 2
Ratio of Areas of inner circle (A1) and circumcircle (A2) = 1 : 2
Hence, the ratio of circumferences of inner circle (A1) and circumcircle (A2) is 1 : √2
& Ratio of Areas of inner circle (A1) and circumcircle (A2) is 1 : 2
HOPE THIS ANSWER WILL HELP YOU….
Answer:
Step-by-step explanation:
Step-by-step explanation:
Given :
Side of a square ABCD = 2a
Diameter of incircle = side of a square = 2a
Radius of a incircle ,r = Diameter of incircle/2
r = 2a/2
r = a
Radius of a incircle ,r = a
Diameter of circumcircle = diagonal of a square = √2 side
Radius of circumcircle ,R = √2 side/2
R = (√2 × 2a) /2
R = √2a
Radius of circumcircle ,R = √2a
(i) Ratio of circumferences of inner circle (C1) and circumcircle (C2)
C1 : C2 = 2πr : 2πR
C1 / C2 = 2πr / 2πR
C1 / C2 = r / R
C1 / C2 = a/√2a
C1 / C2 = 1/√2
C1 : C2 = 1 : √2
Ratio of circumferences of inner circle (C1) and circumcircle (C2) = 1 : √2
(ii) Ratio of Areas of inner circle (A1) and circumcircle (A2) :
A1 : A2 = πr² : πR²
A1 / A2 = πr² / πR²
A1 / A2 = r² /R²
A1 / A2 = a²/(√2a)²
A1 / A2 = a²/2a²
A1 / A2 = ½
A1 : A2 = 1 : 2
Ratio of Areas of inner circle (A1) and circumcircle (A2) = 1 : 2
Hence, the ratio of circumferences of inner circle (A1) and circumcircle (A2) is 1 : √2
& Ratio of Areas of inner circle (A1) and circumcircle (A2) is 1 : 2
HOPE THIS ANSWER HELPED YOU….