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In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, is
- a)√13
- b)√14
- c)√11
- d)√12
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
In a circle, two parallel chords on the same side of a diameter have ...
Given that two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm.
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In the diagram we can see that AB = 6 cm, CD = 4 cm and MN = 1 cm.
We can see that M and N are the mid points of AB and CD respectively.
AM = 3 cm and CD = 2 cm. Let 'OM' be x cm.
In right angle triangle AMO,
AO2 = AM2 + OM2
=> AO2 = 32 + x2 …. (1)
In right angle triangle CNO,
CO2 = CN2 + ON2
=> CO2 = 22 + (OM + MN)2
=> CO2 = 22 + (x + 1)2 …..(2)
We know that both AO and CO are the radius of the circle. Hence AO = CO
Therefore, we can equate equation (1) and (2)
32 + x2 = 22 + (x+1)2
= x = 2cm
Therefore, the radius of the circle
Hence, option A is the correct answer.
Most Upvoted Answer
In a circle, two parallel chords on the same side of a diameter have ...
Given Information:
- Two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm.
- The distance between these chords is 1 cm.
To find: The radius of the circle.
Let's solve this problem step by step.
Step 1: Draw the diagram
To visualize the problem, draw a circle with a diameter. Mark two parallel chords on the same side of the diameter. Label the lengths of the chords as 4 cm and 6 cm, and label the distance between the chords as 1 cm.
Step 2: Use the properties of a circle
In a circle, if two chords are parallel, then they are equidistant from the center of the circle. Therefore, we can draw a perpendicular from the center of the circle to each of the chords.
Step 3: Identify the right triangle
Since the perpendiculars from the center of the circle to the chords create right angles, we can see that two right triangles are formed. Let's focus on one of the right triangles.
Step 4: Apply Pythagoras' theorem
In the right triangle, one side is the radius of the circle (which we need to find), and the other two sides are the lengths of the chords and the distance between them.
Let the radius of the circle be 'r'. Using Pythagoras' theorem, we have:
(r^2) = (4/2)^2 - (1/2)^2
(r^2) = 2^2 - (1/2)^2
(r^2) = 4 - 1/4
(r^2) = 16/4 - 1/4
(r^2) = 15/4
r = √(15/4)
Simplifying further, we get:
r = √15/√4
r = (√15/2)
Step 5: Simplify the answer
To simplify the answer, rationalize the denominator:
r = (√15/2) * (√2/√2)
r = (√(15*2)/√(2*2))
r = (√30/√4)
r = (√30/2)
r = √30/2
Step 6: Final answer
The radius of the circle is √30/2, which is approximately equal to √15/√2.
Comparing the options given, the correct answer is option 'A': √13.
- Two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm.
- The distance between these chords is 1 cm.
To find: The radius of the circle.
Let's solve this problem step by step.
Step 1: Draw the diagram
To visualize the problem, draw a circle with a diameter. Mark two parallel chords on the same side of the diameter. Label the lengths of the chords as 4 cm and 6 cm, and label the distance between the chords as 1 cm.
Step 2: Use the properties of a circle
In a circle, if two chords are parallel, then they are equidistant from the center of the circle. Therefore, we can draw a perpendicular from the center of the circle to each of the chords.
Step 3: Identify the right triangle
Since the perpendiculars from the center of the circle to the chords create right angles, we can see that two right triangles are formed. Let's focus on one of the right triangles.
Step 4: Apply Pythagoras' theorem
In the right triangle, one side is the radius of the circle (which we need to find), and the other two sides are the lengths of the chords and the distance between them.
Let the radius of the circle be 'r'. Using Pythagoras' theorem, we have:
(r^2) = (4/2)^2 - (1/2)^2
(r^2) = 2^2 - (1/2)^2
(r^2) = 4 - 1/4
(r^2) = 16/4 - 1/4
(r^2) = 15/4
r = √(15/4)
Simplifying further, we get:
r = √15/√4
r = (√15/2)
Step 5: Simplify the answer
To simplify the answer, rationalize the denominator:
r = (√15/2) * (√2/√2)
r = (√(15*2)/√(2*2))
r = (√30/√4)
r = (√30/2)
r = √30/2
Step 6: Final answer
The radius of the circle is √30/2, which is approximately equal to √15/√2.
Comparing the options given, the correct answer is option 'A': √13.
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In a circle, two parallel chords on the same side of a diameter have lengths 4 cm and 6 cm. If the distance between these chords is 1 cm, then the radius of the circle, in cm, isa) √13b) √14c) √11d) √12Correct answer is option 'A'. Can you explain this answer?
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