Home
Class 9
MATHS
P Q\ a n d\ R S are two parallel chord...

`P Q\ a n d\ R S` are two parallel chords of a circle and lines `R P\ a n d\ S Q` intersect each other at `O` as shown in Figure. Prove that `O P=O Q`

Text Solution

Verified by Experts

In cyclic quadrilateral
`PQRS`:
`PQ|\|RS`
`=>PR=QS`
We know from the properties of circle,
`OP⋅OR=Hoq⋅OS`
Above can also be written as
`OP⋅(OP+PR)=OQ⋅(OQ+QS)`
`=>OP^2−OQ^2=QS(OQ−OP)`
`=>(OP−OQ)(OP+OQ+QS)=0`
`thereforeOP=OQ`
Hence proved.
Promotional Banner

Topper's Solved these Questions

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    RD SHARMA|Exercise All Questions|205 Videos

  • CONGRUENT TRIANGLE

    RD SHARMA|Exercise All Questions|291 Videos

Similar Questions

Explore conceptually related problems

P Q\ a n d\ R S are two parallel chords of a circle whose centre is O and radius is 10\ c mdot If P Q=16\ c m\ a n d\ R S=12\ c m , find the distance between P Q\ a n d\ R S , if they lie: on the same side of centre O

P Q\ a n d\ R S are two parallel chords of a circle whose centre is O and radius is 10\ c mdot If P Q=16\ c m\ a n d\ R S=12\ c m , find the distance between P Q\ a n d\ R S , if they lie: on opposite side of the centre O

P Q R S is a square such that P R\ a n d\ S Q intersect at Odot State the measure of /_P O Q

l and m are two parallel lines intersected by another pair of parallel lines p and q as shown in figure.Show that o+ABC~=o+CDA Figure

In square A B C D ,\ P\ a n d\ Q are mid-point of A B\ a n d\ C D respectively. If A B=8c m\ a n d\ P Q\ a n d\ B D intersect at O , then find area of O P B

In Figure, P Q R S is a parallelogram, P O\ a n d\ Q O are, respectively, the angle bisectors of /_P\ a n d\ /_Q . Line L O M is drawn parallel to P Q . Prove that: P L=Q M (ii) L O=O M

P Q R S is a parallelogram. P X\ a n d\ Q Y are respectively, the perpendiculars from P\ a n d\ Q to S R\ a n d\ R S produced. Prove that P X=Q Y

In Figure, X\ a n d\ Y are the mid-points of A C\ a n d\ A B respectively, Q P\ B C\ a n d\ C Y Q\ a n d\ B X P are straight lines. Prove that a r\ (\ A B P)=\ a r\ (\ A C Q)

In Figure, /_Q P R=\ /_P Q R\ a n d\ M\ a n d\ N are respectively on sides Q R\ a n d\ P R of P Q R\ such that Q M=P Ndot Prove that O P=O Q , where O is the point of intersection of P M\ a n d\ Q N

A B C D is a parallelogram whose diagonals A C and B D intersect at Odot A Line through O intersects A B at P and D C at Qdot Prove that a r( P O A)=a r( Q O C)dot

RD SHARMA-CIRCLE -All Questions
  1. In Figure, /A=60^0\ a n d\ /A B C=80^0, find /D P C\ a n d\ /B Q C

    Text Solution

    |

  2. In Figure, A B is a diameter of a circle C(O ,\ r)dot Chord C D is ...

    Text Solution

    |

  3. P Q\ a n d\ R S are two parallel chords of a circle and lines R P\ a...

    Text Solution

    |

  4. A B C D is a cyclic quadrilateral whose diagonals A C\ a n d\ B D in...

    Text Solution

    |

  5. P is a point on the side B C of a triangle A B C such that A B=A P...

    Text Solution

    |

  6. A B C is a triangle in which A B=A C\ a n d\ P is a point on A C ....

    Text Solution

    |

  7. In any triangle ABC, if the angle bisector of /Aand perpendicular bis...

    Text Solution

    |

  8. Let the vertex of an angle ABC be located outside a circle and let t...

    Text Solution

    |

  9. In Figure, A B C is an equilateral triangle. Find m\ /B E C

    Text Solution

    |

  10. In Figure, P Q R is an isosceles triangle with P Q=P R\ a n d\ m/P...

    Text Solution

    |

  11. In Figure, O is the centre of the circle. If /B O D=160^0, find the ...

    Text Solution

    |

  12. In Figure A B C D is a cyclic quadrilateral. If /B C D=100^0a n d\ /A ...

    Text Solution

    |

  13. If A B C D is a cyclic quadrilateral in which AD|\|BC . Prove that /...

    Text Solution

    |

  14. In Figure, O is the centre of the circle. Find /C B D

    Text Solution

    |

  15. A B\ a n d\ C D are diameters of a circle with centre Odot If /O B D...

    Text Solution

    |

  16. On a semi-circle with A B as diameter, a point C is taken, so th...

    Text Solution

    |

  17. In a cyclic quadrilateral A B C D, if AB |\| CD\ a n d\ /B=70^@, fin...

    Text Solution

    |

  18. In a cyclic quadrilateral A B C D , if /A=3 /C). Find /A

    Text Solution

    |

  19. In Figure, O is the centre of the circle and /D A B=50^0dot Calculat...

    Text Solution

    |

  20. In Figure, if /B A C=60^0\ a n d\ /B C A=20^0,\ find /A D C

    Text Solution

    |