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21 Oct 2021, 09:00
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
15%
(low)
Question Stats:
83% (02:39) correct
17%
(03:03)
wrong
based on 48
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Square A is inscribed in circle X. Square B is inscribed in circle Y. The perimeter of square B is twice the perimeter of square A. The area of circle Y is equal to k multiplied by the area of circle X. What is the value of k ?
A. 2
B. 3
C. 4
D. 6
E. 8
A. 2
B. 3
C. 4
D. 6
E. 8
Correct Option C = 4
Perimeter square B = (2 * Perimeter square A)
[Lb = 2 * La]
Diameter of circle = Diagonal of square.
r = (d/2) = (√2 L)/2 = (L/√2)
Area of Circle Y = k * Area of circle X
(πr^2)y = k * (πr^2)x
Cancellation of π both side.
(Lb/√2)^2 = k * (La/√2)^2
[Lb = 2 * La]
(2 La/√2)^2 = k * (La/√2)^2
(2 * La^2) = k * (La^2 / 2)
Cancellation La^2 both side
2 = k * (1/2)
k = 4
Option C
Perimeter square B = (2 * Perimeter square A)
[Lb = 2 * La]
Diameter of circle = Diagonal of square.
r = (d/2) = (√2 L)/2 = (L/√2)
Area of Circle Y = k * Area of circle X
(πr^2)y = k * (πr^2)x
Cancellation of π both side.
(Lb/√2)^2 = k * (La/√2)^2
[Lb = 2 * La]
(2 La/√2)^2 = k * (La/√2)^2
(2 * La^2) = k * (La^2 / 2)
Cancellation La^2 both side
2 = k * (1/2)
k = 4
Option C
21 Oct 2021, 10:14
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Method 1:
Rule: given 2 similar figures, if we are given a corresponding dimension of each figure in the ratio of ——> A : B
The area of those 2 similar figures will be in the ratio of ——-> (A)^2 : (B)^2
By virtue of the definition of a circle, every circle is similar to every other circle.
We are given the perimeter of the inscribed square in Circle Y (call this perimeter B) is TWICE the measure of the inscribed square in circle X (call this perimeter A)
Since the square is the largest area quadrilateral that can be drawn within a circle (it is a regular polygon) and every square is similar to every other square, we can use the Perimeter measures as the “corresponding dimension” to determine the Ratio of Area of Circle X - to - Circle Y
A : B = 1 : 2
The Ratio of the Areas of the Circles circumscribing these squares is:
(A)^2 : (B)^2 = (1)^2 : (2)^2 = 1 : 4
Thus:
(Area of Circle Y) = (4) * (Area of Circle X)
K = 4
Answer C
Method 2
Let the perimeter of square B = 8 (each side is 2)
Let the perimeter of square A = 4 (each side is 1)
Rule: the Diagonal of the inscribed square will be equal to the Diameter of the Circle that circumscribes the square
Diagonal of square A = (1) * sqrt(2) = Diameter of circle X
Radius of Circle X = sqrt(2) * (1/2)
Diagonal of Square B = 2 * sqrt(2) = Diameter of circle Y
Radius of Circle Y = sqrt(2)
Area of Circle X = (2/4) (pi) = (1/2) (pi)
Area of Circle Y = (2) (pi)
(2) (pi) = K * (1/2) (pi)
2 = K * (1/2)
K = 4
Answer C
Posted from my mobile device
Rule: given 2 similar figures, if we are given a corresponding dimension of each figure in the ratio of ——> A : B
The area of those 2 similar figures will be in the ratio of ——-> (A)^2 : (B)^2
By virtue of the definition of a circle, every circle is similar to every other circle.
We are given the perimeter of the inscribed square in Circle Y (call this perimeter B) is TWICE the measure of the inscribed square in circle X (call this perimeter A)
Since the square is the largest area quadrilateral that can be drawn within a circle (it is a regular polygon) and every square is similar to every other square, we can use the Perimeter measures as the “corresponding dimension” to determine the Ratio of Area of Circle X - to - Circle Y
A : B = 1 : 2
The Ratio of the Areas of the Circles circumscribing these squares is:
(A)^2 : (B)^2 = (1)^2 : (2)^2 = 1 : 4
Thus:
(Area of Circle Y) = (4) * (Area of Circle X)
K = 4
Answer C
Method 2
Let the perimeter of square B = 8 (each side is 2)
Let the perimeter of square A = 4 (each side is 1)
Rule: the Diagonal of the inscribed square will be equal to the Diameter of the Circle that circumscribes the square
Diagonal of square A = (1) * sqrt(2) = Diameter of circle X
Radius of Circle X = sqrt(2) * (1/2)
Diagonal of Square B = 2 * sqrt(2) = Diameter of circle Y
Radius of Circle Y = sqrt(2)
Area of Circle X = (2/4) (pi) = (1/2) (pi)
Area of Circle Y = (2) (pi)
(2) (pi) = K * (1/2) (pi)
2 = K * (1/2)
K = 4
Answer C
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