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Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. The length of the common chord is
- a)5 cm
- b)8 cm
- c)6 cm
- d)10 cm
Correct answer is option 'C'. Can you explain this answer?
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Two circles of radii 5 cm and 3 cm intersect at two points and the dis...
Given:
- Radii of the circles: 5 cm and 3 cm
- Distance between their centres: 4 cm
To find:
- Length of the common chord
Let's solve this step by step.
1. Draw the circles:
- Draw two circles with radii 5 cm and 3 cm.
- Mark their centres as A and B.
- Draw a line segment AB to represent the distance between their centres, which is 4 cm.
2. Draw the common chord:
- Draw a line segment CD such that it intersects both circles at points E and F.
- Let's assume that E lies on the circle with radius 5 cm and F lies on the circle with radius 3 cm.
3. Find the lengths CE and FD:
- CE = radius of the larger circle - distance between the centres
= 5 cm - 4 cm
= 1 cm
- FD = radius of the smaller circle - distance between the centres
= 3 cm - 4 cm
= -1 cm (negative value since it lies inside the circle)
4. Find the lengths AE and BF:
- AE = radius of the larger circle
= 5 cm
- BF = radius of the smaller circle
= 3 cm
5. Apply the Pythagorean theorem:
- In triangle AEC, AE^2 = AC^2 + CE^2
=> 5^2 = AC^2 + 1^2
=> AC^2 = 25 - 1
=> AC^2 = 24
=> AC = √24 cm
=> AC = 2√6 cm (approx. 4.90 cm)
- In triangle BFD, BF^2 = BD^2 + FD^2
=> 3^2 = BD^2 + (-1)^2
=> BD^2 = 9 - 1
=> BD^2 = 8
=> BD = √8 cm
=> BD = 2√2 cm (approx. 2.83 cm)
6. Length of the common chord:
- Length of the common chord = AC + BD
= 2√6 cm + 2√2 cm
= 2(√6 + √2) cm (approx. 7.73 cm)
Therefore, the length of the common chord is approximately 7.73 cm, which is closest to option 'C' (6 cm).
- Radii of the circles: 5 cm and 3 cm
- Distance between their centres: 4 cm
To find:
- Length of the common chord
Let's solve this step by step.
1. Draw the circles:
- Draw two circles with radii 5 cm and 3 cm.
- Mark their centres as A and B.
- Draw a line segment AB to represent the distance between their centres, which is 4 cm.
2. Draw the common chord:
- Draw a line segment CD such that it intersects both circles at points E and F.
- Let's assume that E lies on the circle with radius 5 cm and F lies on the circle with radius 3 cm.
3. Find the lengths CE and FD:
- CE = radius of the larger circle - distance between the centres
= 5 cm - 4 cm
= 1 cm
- FD = radius of the smaller circle - distance between the centres
= 3 cm - 4 cm
= -1 cm (negative value since it lies inside the circle)
4. Find the lengths AE and BF:
- AE = radius of the larger circle
= 5 cm
- BF = radius of the smaller circle
= 3 cm
5. Apply the Pythagorean theorem:
- In triangle AEC, AE^2 = AC^2 + CE^2
=> 5^2 = AC^2 + 1^2
=> AC^2 = 25 - 1
=> AC^2 = 24
=> AC = √24 cm
=> AC = 2√6 cm (approx. 4.90 cm)
- In triangle BFD, BF^2 = BD^2 + FD^2
=> 3^2 = BD^2 + (-1)^2
=> BD^2 = 9 - 1
=> BD^2 = 8
=> BD = √8 cm
=> BD = 2√2 cm (approx. 2.83 cm)
6. Length of the common chord:
- Length of the common chord = AC + BD
= 2√6 cm + 2√2 cm
= 2(√6 + √2) cm (approx. 7.73 cm)
Therefore, the length of the common chord is approximately 7.73 cm, which is closest to option 'C' (6 cm).
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Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. The length of the common chord isa)5 cmb)8 cmc)6 cmd)10 cmCorrect answer is option 'C'. Can you explain this answer?
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