Two concentric circles are given. Prove that all chords of the circle with the larger
radius which touch the circle with the smaller radius are congruent.

Asked by shailesh.solanki | 18th Mar, 2018, 07:10: PM

Expert Answer:

begin mathsize 16px style Given colon
Two space concentric space circles space with space centres space at space straight O. space
AB space and space CD space be space two space chords space of space the space outer space circle
which space touch space the space inner space circle space at space the space points space straight P space and space straight Q space respectively.
To space prove colon space radius space of space outer space circle equals radius space of space inner space circle.
Proof colon
It space is space sufficient space to space prove space AB equals CD.
Join space OP comma space OQ comma space OB comma space OD
Let space straight R space be space the space radius space of space outer space circle space and space straight r space be space the space radius space of space inner space circle.
CD space touches space inner space circle space at space straight Q.
CD space is space tangent space to space inner space circle.
OQ perpendicular CD
QD equals 1 half CD
CD equals 2 QD
OP perpendicular AB
AB equals 2 PB
In space increment OQD comma
OQ squared plus QD squared equals OD squared
QD squared equals straight R squared minus straight r squared.............. left parenthesis straight i right parenthesis
In space increment space OPB comma
OP squared plus PB squared equals OB squared
PB squared equals OB squared minus OP squared
PB squared equals straight R squared minus straight r squared................ left parenthesis ii right parenthesis
QD squared equals PB squared................ From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis
QD equals PB
CD equals 2 QD space and space AB equals 2 PB
CD equals 2 square root of straight R squared minus straight r squared end root space and space AB equals 2 square root of straight R squared minus straight r squared end root space
AB equals CD end style

Answered by Sneha shidid | 19th Mar, 2018, 10:36: AM

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