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Q and R are the centers of two congruent circles intersecting each other at points C and D. The line joining their centers intersects the circle in points A and B such that A and B do not lie between Q and R. If CD = 6 cm and AB = 12 cm, determine the radius of each circle and the distance between the centers of two circles.

Asked by gpnkumar0 7th March 2017, 6:58 PM
Answered by Expert
Answer:
begin mathsize 12px style Given space that space the space circles space are space congruent.
So comma space AQ equals QC equals QD equals RB equals RC equals RD equals straight r space left parenthesis say right parenthesis
SinceQC equals QD equals RC equals RD
rightwards double arrow QCRD space is space straight a space rhombus
Since space diagonals space of space straight a space rhombus space are space perpendicular space bisectors space of space eachother comma
so comma space CD perpendicular QR space and space CD space bisects space QR
that space is comma space QP equals RP space and space CP equals PD
CD equals 6 space cm rightwards double arrow CP equals CD over 2 equals 3 space cm
AB equals 12 space cm space
and space AB equals AQ plus QR plus QB
rightwards double arrow QP equals QR over 2
equals AB equals AQ plus QR plus RB
rightwards double arrow QR equals AB minus AQ minus RB
rightwards double arrow QR equals 12 minus straight r minus straight r equals 12 minus 2 straight r equals 2 left parenthesis 6 minus straight r right parenthesis
So comma space QP equals fraction numerator 2 left parenthesis 6 minus straight r right parenthesis over denominator 2 end fraction equals left parenthesis 6 minus straight r right parenthesis space cm
In space right space increment QPC comma
QC squared equals CP squared plus OP squared space.... left parenthesis Pythagoras space theorem right parenthesis
rightwards double arrow QC squared equals 3 squared plus left parenthesis 6 minus straight r right parenthesis squared
In space right space increment QPC comma
QC squared equals CP squared plus QP squared space.... left parenthesis Pythagoras space theorem right parenthesis
rightwards double arrow straight r squared equals 3 squared plus left parenthesis 6 minus straight r right parenthesis squared
rightwards double arrow straight r squared equals 3 squared plus 36 minus 12 straight r plus straight r squared
rightwards double arrow 45 equals 12 straight r
rightwards double arrow straight r equals 3.75 space cm end style
Answered by Expert 8th March 2017, 8:52 AM
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