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Two circles of radius 37 cm and 20 cm intersect each other at P, and Q. O and O’ are the centres of the circles. If the length of PQ is 24 cm, then the distance between their centre is:
- a)35 cm
- b)51 cm
- c)21 cm
- d)16 cm
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Two circles of radius 37 cm and 20 cm intersect each other at P, and ...
OP= 37 cm and O'P = 20 cm
PQ = 24 cm
PM = MQ = 12 cm
O'M = 16 cm
OO' = OM + 0'M
OO' = 35 + 16 = 51 cm
Most Upvoted Answer
Two circles of radius 37 cm and 20 cm intersect each other at P, and ...
Understanding the Problem
To find the distance between the centers O and O' of the two intersecting circles, we can use the properties of intersecting circles. We know the following:
- Radius of Circle 1 (O): 37 cm
- Radius of Circle 2 (O’): 20 cm
- Length of chord PQ (intersection points): 24 cm
Using the Chord Length Formula
The distance between the centers (d) can be calculated using the relation involving the radius and the length of the chord. For two intersecting circles, the following equation holds:
d^2 = r1^2 + r2^2 - 2 * r1 * r2 * cos(θ)
where r1 and r2 are the radii, and θ is the angle subtended by the chord at the centers. However, since we have the length of the chord, we can use a simpler geometric approach.
Applying the Right Triangle Method
1. Half of Chord Length: PQ = 24 cm, so half of PQ = 12 cm.
2. Distances to the Midpoint: Let M be the midpoint of PQ. The distances from O to M and O' to M can be derived using the Pythagorean theorem.
- For Circle O (radius = 37 cm):
OM = √(r1² - (PQ/2)²) = √(37² - 12²)
- For Circle O’ (radius = 20 cm):
O'M = √(r2² - (PQ/2)²) = √(20² - 12²)
3. Distance Between Centers: The total distance d = OM + O'M.
Calculating Distances
- For Circle O:
OM = √(1369 - 144) = √1225 = 35 cm
- For Circle O’:
O'M = √(400 - 144) = √256 = 16 cm
Thus, the total distance (d):
d = OM + O'M = 35 cm + 16 cm = 51 cm.
Conclusion
Therefore, the distance between the centers O and O' is 51 cm, making option 'B' the correct answer.
To find the distance between the centers O and O' of the two intersecting circles, we can use the properties of intersecting circles. We know the following:
- Radius of Circle 1 (O): 37 cm
- Radius of Circle 2 (O’): 20 cm
- Length of chord PQ (intersection points): 24 cm
Using the Chord Length Formula
The distance between the centers (d) can be calculated using the relation involving the radius and the length of the chord. For two intersecting circles, the following equation holds:
d^2 = r1^2 + r2^2 - 2 * r1 * r2 * cos(θ)
where r1 and r2 are the radii, and θ is the angle subtended by the chord at the centers. However, since we have the length of the chord, we can use a simpler geometric approach.
Applying the Right Triangle Method
1. Half of Chord Length: PQ = 24 cm, so half of PQ = 12 cm.
2. Distances to the Midpoint: Let M be the midpoint of PQ. The distances from O to M and O' to M can be derived using the Pythagorean theorem.
- For Circle O (radius = 37 cm):
OM = √(r1² - (PQ/2)²) = √(37² - 12²)
- For Circle O’ (radius = 20 cm):
O'M = √(r2² - (PQ/2)²) = √(20² - 12²)
3. Distance Between Centers: The total distance d = OM + O'M.
Calculating Distances
- For Circle O:
OM = √(1369 - 144) = √1225 = 35 cm
- For Circle O’:
O'M = √(400 - 144) = √256 = 16 cm
Thus, the total distance (d):
d = OM + O'M = 35 cm + 16 cm = 51 cm.
Conclusion
Therefore, the distance between the centers O and O' is 51 cm, making option 'B' the correct answer.
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Two circles of radius 37 cm and 20 cm intersect each other at P, and Q. O and O’ are the centres of the circles. If the length of PQ is 24 cm, then the distance between their centre is:a)35 cmb)51 cmc)21 cmd)16 cmCorrect answer is option 'B'. Can you explain this answer?
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