Question 1: A chord of length 6 \ cm is drawn in a circle of radius 5 \ cm . Calculate its distance from the center of the circle.

Answer:

Let the distance from the center = x

Therefore x = \sqrt{5^2-3^2}= \sqrt{25-9} = 4 \ cm 

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Question 2: A chord of length 8 \ cm is drawn  at a distance of 3 \ cm from the center of a circle. Calculate the radius of the circle.

Answer:

Let the radius = r \ cm

Therefore r = \sqrt{4^2+3^2} = 5 \ cm 

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Question 3: The radius of a circle is 17 \ cm and the length of perpendicular drawn from the center to a chord is 8 \ cm . Calculate the length of the chord.

Answer:

Let the length of the chord = 2x

Therefore x = \sqrt{17^2-8^2} =\sqrt{289-64} = 15 \ cm

Therefore the length of the chord = 2x = 30 \ cm

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Question 4: A chord of length 24 cm is at a distance of 5 cm from the center of the circle. Find the length of the chord of the same circle which is at a distance of 12 cm from the center.

Answer:

Let the radius = r \ cm

Therefore r = \sqrt{12^2+5^2} = 13 \ cm 

Let the length of the chord = 2x

Therefore x = \sqrt{13^2-12^2} =\sqrt{169-144} = 5 \ cm

Therefore the length of the chord = 2x = 10 \ cm

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c1Question 5: In the following figure, AD is a straight line OP \perp AD and O is the center of both the circles. If OA = 34 \ cm, OB = 20 \ cm and OP = 16 \ cm , find the length of AB .

Answer:

BP = \sqrt{20^2-16^2}=\sqrt{400-256}= 12 \ cm

AP = \sqrt{34^2-16^2}=\sqrt{1156-256}= 30 \ cm

Therefore AB = (AP-BP) = 30 - 12 = 18 \ cm

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Question 6: O is the center of the circle of radius 10 \ cm . P is any point in the circle such that OP=6 \ cm . A is the point travelling along the circumference, x is the distance from A to P . What are the least and the greatest values of x \ in \ cm . What is the position of the points O, P \ and \ A at these values . [1992]

Answer:

When P is on OA , the least distance of x = 4 \ cm

When P is on extended OA , the greatest distance of x = 16 \ cm 

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Question 7: In a circle of radius 17 \ cm  , two parallel chords of length 30 \ cm  and 16 \ cm  are drawn. Find the distance between the chords, if both the chord are:

(i) on the opposite side of the center

(ii) on the same side of the center

Answer:

(i) When the chords are on the opposite side of the triangles

Distance of the larger chord from the center x_1 = \sqrt{17^2-15^2}= 8 \ cm 

Distance of the smaller chord from the center x_2 = \sqrt{17^2-8^2}= 15 \ cm 

Therefore the distance between the chords = x_1 + x_2 = 23 \ cm 

(ii)  When the chords are on the same side of the triangles

Distance of the larger chord from the center x_1 = \sqrt{17^2-15^2}= 8 \ cm 

Distance of the smaller chord from the center x_2 = \sqrt{17^2-8^2}= 15 \ cm 

Therefore the distance between the chords = x_2 - x_1 = 7 \ cm 

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Question 8: Two parallel chords are drawn in a circle of diameter 30 \ cm  . The length of one chord is 24 \ cm  . and the distance between the two chords is 21 \ cm  . Find the length of another chord.

Answer:

Let the distance of the 24 \ cm  chord from the center = x \ cm 

Therefore x = \sqrt{15^2-12^2}= 9 \ cm 

Therefore the distance of the other chord from the center = 21 - 9 = 12 \ cm 

Therefore the length of the other chord = 2 \times \sqrt{15^2-12^2} = 18 \ cm 

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c2Question 9: A chord CD of a circle whose center is O , is bisected at P by a diameter AB . Given  OA = OB = 15 \ cm and OP = 9 \ cm , calculate the lengths of  (i) CD (ii) AD and (iii) CB

Answer:

CP = \sqrt{15^2-9^2} = 12 \ cm

CD = 2 \times 12 = 24 \ cm

AD = \sqrt{24^2+12^2}=\sqrt{720} = 26.83 \ cm

CB = \sqrt{12^2+(15-9)^2} = \sqrt{144+36} = 13.42 \ cm

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c3Question 10: The figure given below, shows a circle with center O in which Diameter AB bisects the chord CD at point E . If CE=ED=8 \ cm and EB=4 \ cm , find the radius of the circle.

Answer:

Let OE = x \ cm

Therefore

x^2+8^2 = (x+4)^2

x^2 + 64 = x^2 + 8x + 16

8x = 48 \Rightarrow x = 6 \ cm

Therefore Radius = 6 + 4 = 10 \ cm

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