In the given figure, the radii of two concentric circles are 12 cm12 cm and 7 cm7 cm. ABAB is the diameter of the bigger circle and BDBD is a tangent to the smaller circle touching it at DD. Find the length ADAD.
D O B E A


Answer:

17.06 cm17.06 cm

Step by Step Explanation:
  1. We know that angle in a semicircle is of 90.90. So, AEB=90.AEB=90.
  2. We also know that in two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
    So, ODBEODBE and ODOD bisects BEBE.
  3. Using Pythagoras' theorem in right OBDOBD, we have OB2=OD2+BD2OB2=OD2+BD2 It is given that OB=12 cmOB=12 cm and OD=7 cm.OD=7 cm. BD=OB2OD2=(12)2(7)2 cm=95 cmBD=OB2OD2=(12)2(7)2 cm=95 cm Now, BE=2BD=295 cm.    [ D is the midpoint of BE ] BE=2BD=295 cm.    [ D is the midpoint of BE ] 
  4. Using Pythagoras' theorem in right AEBAEB, we have AB2=AE2+BE2 As, AB is the diameter of the circle, AB = 2×OB=2×12 cm=24 cm AE=AB2BE2=(24)2(295)2 cm=196 cm
  5. Using Pythagoras' theorem in right AED, we have AD2=AE2+DE2 We know that DE=BD=95 cm   [As, OD bisects BE]

    AD=AE2+DE2=(196)2+(95)2 cm=17.06 cm

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