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Practice each question, then open the answer or worked solution.
If the sum of the areas of two circles with radii R1 and R2 is equal to the area of a circle of radius R, then
(A) R1 + R2 = R (B) R12 + R22 = R2
(C) R1 + R2 < R (D) R12 + R22 < R2
(B) R12 + R22 = R2
Explanation:
According to the question,
Area of circle = Area of first circle + Area of second circle
∴ πR2 = πR12 + πR22
⇒ R2 = R12 + R22
∴ Option B is correct.
If the sum of the circumferences of two circles with radii R1 and R2 is equal to the circumference of a circle of radius R, then
(A) R1 + R2 = R (B) R1 + R2 > R
(C) R1 + R2 < R (D) Nothing definite can be said about the relation among R1, R2 & R.
Explanation:
According to the question,
Circumference of circle with radius R = Circumference of first circle with radius R1 + Circumference of second circle with radius R2
∴ 2πR = 2πR1+ 2πR2
⇒ R = R1+ R2
∴ Option A is correct.
If the circumference of a circle and the perimeter of a square are equal, then
(A) Area of the circle = Area of the square
(B) Area of the circle > Area of the square
(C) Area of the circle < Area of the square
(D) Nothing definite can be said about the relation between the areas of the circle & square.
(B) Area of the circle > Area of the square
Explanation:
According to the question,
Circumference of a circle = Perimeter of square
Let r be the radius of the circle and a be the side of square.
∴ From the given condition, we have 2π r = 4a
(22/7)r = 2a
⇒ 11r = 7a
⇒ a = (11/7)a
⇒ r = (7/11)a …………..(i)
Now, area of circle = A1 = πr2 and area of square = A2 = a2
From equation (i ), we have
A1 = π × (7/11)2
= (22/7) × (49/121)a2
= (14/11)a2 and A2 = a2
∴ A1 = (14/11) A2
⇒ A1 > A2
Hence, Area of the circle > Area of the square.
∴ Option B is correct.
Area of the largest triangle that can be inscribed in a semi-circle of radius r units is
(A) r2 sq. units (B) ½ r2 sq. units
(C) 2 r2 sq. units (D) √2 r2 sq. units
(A) r2 sq. units
Explanation:
The largest triangle that can be inscribed in a semi-circle of radius r units is the triangle having its base as the diameter of the semi-circle and the two other sides are taken by considering a point C on the circumference of the semi-circle and joining it by the end points of diameter A and B.
∴ ∠ C = 90° (by the properties of circle)
So, ΔABC is right angled triangle with base as diameter AB of the circle and height be CD.
Height of the triangle = r
∴ Area of largest ΔABC = (1/2)× Base × Height = (1/2)× AB × CD
= (1/2)× 2r × r = r2 sq. units
∴ Option A is correct.
If the perimeter of a circle is equal to that of a square, then the ratio of their areas is
(A) 22 : 7 (B) 14 : 11 (C) 7 : 22 (D) 11: 14
(B) 14 : 11
Explanation:
Let r be the radius of the circle and a be the side of the square.
According to the question,
Perimeter of a circle = Perimeter of a square
⇒ 2πr = 4a
⇒ a = πr/2
Area of the circle = r2 and Area of the square = a2
Now, Ratio of their areas = (Area of circle)/(Area of square)
= 4/π
= [4/(22/7)]
= 14/11
∴ Option B is correct.