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Prove that, of any two chords of a circle, the greater chord is nearer to the centre.
Solution
Given: AB and CD are two chords of a circle C(O, r). OP < OQ, where OP ⊥ CD and OQ ⊥ AB To prove: CD > AB Proof:
AQ = 12 AB (Perpendicular from the centre of a circle to a chord bisects the cord)
CP = 12 CD (Perpendicular from the centre of a circle to a chord bisects the cord) In right ΔAOQ,
⇒ AO2 = OQ2 + AQ2
⇒ AQ2 = AO2 - OQ2 ....(1) In right ΔCOP,
⇒ CO2 = CP2 + OP2
⇒ CP2 = CO2 - OP2 ....(2) ⇒ OP < OQ
⇒ OP2 < OQ2
⇒ -OP2 < -OQ2
⇒ CO2 - OP2 < AO2 - OQ2 (CO = AO)
⇒ CP2 > AQ2 (from (1) and (2))
⇒14 CD2 > 14 AB2
⇒ CD2 > AB2
⇒ CD > AB
Mathematics
Concise Mathematics
Standard X
3
Given: A circle with centre O and radius r.OM ⊥ AB and ON ⊥ CD Also AB > CDTo prove: OM < ONProof: Join OA and OC.In Rt. ΔAOM, `AO^2 = AM^2 + OM^2``r^2 = (1/2 AB)^2 + OM^2 ``r^2 = 1/4 AB^2 + OM^2` ………..(i)Again in Rt. ΔONC,`OC^2 = NC^2 + ON^2`
`⇒ r^2 = (1/2 CD )^2 + ON^2`
` ⇒ r^2 = 1/4 CD^2 + ON^2` ………..(ii)From (i) and (ii)`1/4 AB^2 + OM^2 = 1/4 CD^2 + ON^2`But, AB > CD (given)∴ ON > OM⇒ OM < ON
Hence, AB is nearer to the centre than CD.