Concentric Circles Problems


 
 
Concept Explanation
 

Concentric Circles Problems

Concentric Circles: The circle which have common centre point and different radius are called concentric circles.

Illustration:   Two concentric circles with centre O have, A, B, C, D as the points of intersection with the line l as shown in the diagram. If AD = 12 cm and BC = 8 cm, find the lengths of AB, CD, AC and BD.

SOLUTION  Since large OM perp BC

large therefore ;;BM = CM = frac{1}{2}BC = 4cm

Similarly, large OM perp AD

large Rightarrow AM = DM = frac{1}{2}AD=6cm.

Now,   large AB = AM - BM = (6-4)cm = 2cm

Also,  large CD = DM-CM =(6-4)cm = 2cm

large therefore AC=AB+BC=(2+8)cm=10cm

and, large BD =BC+CD=(8+2)cm=10cm

Illustration:   Two concentric circles C_1 and C_2 with centre O have, A, B, C, D as the points of intersection with the line l as shown in the diagram. If AD = 8 cm and BC = 6 cm, and the distance of the chord from centre is 2 cm,find the length of radius of two circles.

SOLUTION  Since large OM perp BC

large therefore ;;BM = CM = frac{1}{2}BC = frac{6}{2}=3cm

Similarly, large OM perp AD

large Rightarrow AM = DM = frac{1}{2}AD=frac{8}{2}= 4cm.

Now  large Delta AOM is a right angled at M

large OA^2=OM^2+AM^2       [ By pythagorous Theorem]

large OA^2=2^2+4^2= 4 +16 = 20

large OA=sqrt {20} =2sqrt 5 cm

Now  large DeltaBOM is a right angled at M

large OB^2=OM^2+BM^2       [ By pythagorous Theorem]

large OB^2=2^2+3^2= 4 +9 = 13

large OB=sqrt {13} cm

Hence radius of circle large C_1; is;sqrt {13} cm; and;C_2; is;2sqrt {5} cm;

 

Sample Questions
(More Questions for each concept available in Login)
Question : 1

The figure shows two concentric circles and AD is a chord of larger circle. If the length of AP = 10 cm and BP = 3cm.  Find the value of 3CD + 2AP

 

Right Option : A
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Explanation
Question : 2

The figure shows two concentric circles and AD is a chord of larger circle. If the length of AP = 10 cm and BP = 3cm.  Find the value of CD.

 

Right Option : D
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Explanation
Question : 3

If radii of two concentric circles are 4 cm and 5 cm, then the length of the chord of one circle which is tangent to the other circle is 

Right Option : B
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Explanation
 
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