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Two chords AB and AC of a circle subtend...

Two chords AB and AC of a circle subtends angles equal to `90^(@) and 150^(@)`, respectively at the centre. Find`angleBAC`, if AB and AC lie on the opposite sides of the centre.

Text Solution

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In `DeltaBOA, OB=OA` [both are the radius of circle]
`:. angleOAB=angleOBA …(i)`
[angles opposite to equal sides are equal]

In `DeltaOAB, angleOBA+angleOAB+angleAOB=180^(@)` [by angle sum property of a triangle]
`rArr angleOAB+angleOAB+90^(@)=180^(@)` [from Eq. (i)]
`rArr 2angleOAB=180^(@)-90^(@)`
`rArr angleOAB=90^(@)/2=45^(@)`
Now, in `DeltaAOC, AO=OC` [both are the redius of a circle]
`:. angleOCA=angleOAC`...(ii)
[angles opposite to equal sides are equal]
Also, `angleAOC+angleOAC+angleOCA=180^(@)` [by angle sum property of a triangle]
`rArr 150^(@)+2 angleOAC=180^(@)` [from Eq. (ii)]
`rArr 2angleOAC=180^(@)-150^(@)`
`rArr 2 angle OAC=30^(@)`
`rArr angleOAC=15^(@)`
`:. angleBAC=angleOAB+angleOAC=45^(@)+15^(@)=60^(@)`
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