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If O is the circumcentre of a A B C and...

If `O` is the circumcentre of a ` A B C` and `O D_|_B C ,` prove that `/_B O D=/_Adot`

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We conclude from the given information
`O` is the circum centre of `/_\ABC`
`OD_|_BC`
`OB` is joined.
Construction :
Join `OC`.
Proof : Arc BC subtends `/_BOC` at the centre and `/_BAC` at the remaining part at the circle.`/_BOC=2/_A. .........(i)`
In right
`/_\OBD and /_\OCD`
Side `OD = OD` (Common)
Hyp. `OB = OC` (Radii of the circle)
`:./_\OBD=/_\OCD` (RHS criteria)
`:.BOD=/_COD=1/2/_BOC`
`=>/_BOC=2/_BOD ......(ii)`
From `(i)` and `(ii)`
`2 /_BOD=2/_A`
`:./_BOD=/_A`
Hence proved.
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RD SHARMA-CIRCLE -All Questions
  1. In Figure, O is the centre of the circle. Find /B A C

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  2. If O is the center of the circle, find the value of x in each of t...

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  3. If O is the circumcentre of a A B C and O D|B C , prove that /B O D=/...

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  4. In the figure, O is the center of the circle and BO is the bisector of...

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  5. In Figure, O is the centre of the circle, prove that /x=/y+/z

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  6. In Figure, O\ a n d\ O ' are centres of two circles intersecting at...

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  7. In Figure, O is the centre of a circle and P Q is a diameter. If /R...

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  8. In Figure, if /A C B=40^0,\ /D P B=120^0,\ find /C B D

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  9. A chord of a circle is equal to the radius of the circle. Find the a...

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  10. In Figure, P Q R S is a cyclic quadrilateral. Find the measure of each...

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  11. In Figure, if /D B C=70^0\ a n d\ /B A C=30^0,\ find /B C Ddot Fur...

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  12. In Figure, A B C D is a cyclic quadrilateral; O is the centre of th...

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  13. In Figure, A B C is an isosceles triangle with A B=A C\ a n d\ m\ ...

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  14. In Figure, A B C D is a cyclic quadrilateral whose side A B is a ...

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  15. In Figure, C\ a n d\ D are points on the semi-circle described on B...

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  16. In Figure, O is the centre of the circle. The angle subtended by the a...

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  17. In Figure, B D=D C\ a n d\ /D B C=25^0 find the measure of /B A C

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  18. Prove that any cyclic parallelogram is a rectangle.

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  19. If diagonals of a cyclic quadrilateral are diameters of the circle t...

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  20. Two circles intersect at two points B and C. Through B, two line seg...

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