Home
Class 9
MATHS
If the bisectors of the opposite angles ...

If the bisectors of the opposite angles `/_Pa n d/_R` of a cyclic quadrilateral `P Q R S` intersect the corresponding cicle at `Aa n dB` respectively, then `A B` is a diameter of the circle.

Text Solution

Verified by Experts

`/_P+/_R=180^@`
`1/2/_P+1/2/_R=90^@`
`/_APS+/_BRS=90^@-(1)`
`/_BRS=/_BPS-(2)`
`/_APs+/_BPS=90^@`
`/_APB=90^@`
AB is diameter of circle.
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

If the bisectors of the opposite angle /_P and /_Rof a Cyclic quadrilateral PQRS intersect the cooresponding circle at A and B respectively; then AB is a diameter of the circle.

If bisectors of opposite angles of a cyclic quadrilateral ABCD intersect the circle, circumscribing it at the points P and Q, prove that PQ is a diameter of the circle.

If the bisectors of two adjacent angles A\ a n d\ B of a quadrilateral A B C D intersect at a point O such that /_C+/_D=k/_A O B , then find the value of k

The bisectors of the angles formed by producing the opposite sides of a cyclic quadrilateral (provided that they are not parallel) intersect at right angle.

In a cyclic quadrilateral A B C D if A B C D a n d /_B=70^0, find the remaining angles.

ABCD is a cyclic quadrilateral. The bisectors of the angles A, B, C and D. cut the circle at P, Q, R and S respectively. What is anglePQR + angleRSP equal to ?

If bisectors of angleA and angleB of a quadrilateral ABCD intersect each other at P, of angleB and angleC at Q, of angleC and angleD at R and of angleD and angleA at S, then PQRS is a

If a,b in N , PQRS is a square and P, Q, R, S are mid points of AB, AD, CD, BC respectively then a+b is

RD SHARMA-CIRCLE -All Questions
  1. If the sum of any pair of opposite angles of a quadrilateral is 180^@;...

    Text Solution

    |

  2. The sum of the angles in the four segments exterior to a cyclic qua...

    Text Solution

    |

  3. If the bisectors of the opposite angles /Pa n d/R of a cyclic quadrila...

    Text Solution

    |

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

    Text Solution

    |

  5. Two diameters of a circle intersect each other at right angles. Prove ...

    Text Solution

    |

  6. A B C and A D C are two right triangles with common hypotenuse A Cdot ...

    Text Solution

    |

  7. In a circle with centre O , chords A Ba n dC D intersect inside the ci...

    Text Solution

    |

  8. triangle ABC AND triangle ADC are two right triangles with common hypo...

    Text Solution

    |

  9. Bisectors of angles A, B and C of a triangle ABC intersect its circ...

    Text Solution

    |

  10. Prove that the circle drawn with any side of a rhombus as a diameter, ...

    Text Solution

    |

  11. A Ca n dB D are chords of a circle that bisect each other. Prove that:...

    Text Solution

    |

  12. Prove that the mid-point of the hypotenuse of a right triangle is eq...

    Text Solution

    |

  13. In Figure, P , is any point on the chord B C of a circle such that A B...

    Text Solution

    |

  14. Two chords A B and C D of a circle are parallel and a line l is the pe...

    Text Solution

    |

  15. A circular park of radius 20m is situated in a colony. Three boys Anku...

    Text Solution

    |

  16. In an equilateral triangle, prove that the centroid and the circumcent...

    Text Solution

    |

  17. In Figure, l is a line intersecting the two concentric circles, whose ...

    Text Solution

    |

  18. In the Figure, O D is perpendicular to the chord A B of a circle who...

    Text Solution

    |

  19. AB and CD are two chords of a circle such that AB = 6 cm, CD = 12 cm a...

    Text Solution

    |

  20. A B and C D are two parallel chords of a circle such that A B=10c m an...

    Text Solution

    |