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Question
Draw two circles of radii 3.5 cm and 2 cm respectively so that their centres are 6 cm apart. Draw direct common tangents to the circle and show that they are equal in length.
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Solution
Steps of construction:
(i) Draw a line OP= 6 cm.
(ii) At O, draw a circle of radius 3.5 cm.
(iii) At P, draw a circle of radius 2 cm.
(iv) At O, draw a third circle concentric to the bigger circle and radius = (3.5 - 2) cm= 1.5 cm
(v) Draw a perpendicular bisector of OP. Let R be the mid-point of OP.
(vi) With R as centre and OR as radii, draw a fourth circle. Mark as T and S where the third and fourth circles intersect each other.
(vii) Join OT and OS and extend lines to meet the bigger cirde at A and B.
(viii) Join PT and PS.
(ix) On PT and PS, draw perpendiculars to meet the smaller circle at Mand N.
(x) Join AM and BN.
AM and BN are the required tangents.
Proof:
Since AT || PM and BS || PN; therefore AM = PT and BN = PS
Now in Δ OTP and Δ OSP
PT = PS (Tangents to a circle from same point)
Therefore, AM = BN
Hence, proved.
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