Home
Class 9
MATHS
In a triangle A B C median A D is produc...

In a triangle `A B C` median `A D` is produced to `X` such that `A D=D Xdot` Prove that `A B X C` is a parallelogram.

Text Solution

Verified by Experts

Given : In `/_\ABC, AD` is median and `AD` is produced to `X` such that `DX=AD`
To prove : `ABXC` is a parallelogram.
Construction : Join `BX` and `CX`.
Proof : In `/_\ABD and /_\CDX`,
`AD=DX` (Given)
`BD=DC` (`D` is mid-points)
`/_ADB=/_CDX` (Vertically opposite angles)
`:./_\ABD~=/_\CDX` (By `SAS`)
`:. AB=CX` (By `CPCT`)
and `/_ABD=/_DCX`
But these are alternate angles.
`:.AB||CX` and `AB=CX`
`:. ABXC` is a parallelogram.
Promotional Banner

Topper's Solved these Questions

  • PROBABILITY

    RD SHARMA|Exercise All Questions|67 Videos

  • RATIONALISATION

    RD SHARMA|Exercise All Questions|130 Videos

Similar Questions

Explore conceptually related problems

In a A B C median A D is produced to X such that A D=D X . Prove that A B X C is a parallelogram.

A B C D is a parallelogram. P is a point on A D such that A P=1/3\ A D\ a n d\ Q is a point on B C such that: C Q=1/3B Cdot Prove that A Q C P is a parallelogram.

A B C D is a parallelogram. E is a point on B A such that B E=2\ E A\ a n d\ F is a point on D C such that D F=2\ F Cdot Prove that A E\ C F is a parallelogram whose area is one third of the area of parallelogram A B\ C Ddot

The sides A B\ a n d\ C D of a parallelogram A B C D are bisected at E\ a n d\ Fdot Prove that E B F D is a parallelogram.

In Figure, A B C D is a parallelogram and X\ a n d\ Y are points on the diagonal B D such that D X=B Y . Prove that A X C Y is a parallelogram (ii) A X=C Y ,\ A Y=C X A Y B\ ~= C X D (iv) A X D~=\ C Y B

Any point D is taken in the base B C of a triangle A B C\ a n d\ A D is produced to E , making D E equal to A Ddot Show that a r\ ( B C E)=a r( A B C)

Delta ABC is an isosceles triangle in which A B= A C . Side BA is produced to D such that A D = A B (see Fig. 7.34). Show that /_B C D is a right angle.

In Figure, A B C D is a trapezium in which A B||D C ,\ \ D C is produced to E such that C E=A B , prove that a r( A B D)=a r\ ( B C E)dot Construction: Draw D M\ on B A produced and B N_|_D C

In a parallelogram A B C D , the bisector of /_A also bisects B C at Xdot Prove that A D=2A B

In figure A B=A C\ a n d\ C P||B A and A P is the bisector of exterior /_C A D of A B C . Prove that /_P A C=/_B C A (ii) A B C P is a parallelogram

RD SHARMA-QUADRILATERALS-All Questions
  1. A quadrilateral is a parallelogram, if its one pair of opposite sides ...

    Text Solution

    |

  2. In Figure, A B C D is a parallelogram and X ,Y are the mid=points of s...

    Text Solution

    |

  3. In a triangle A B C median A D is produced to X such that A D=D Xdot P...

    Text Solution

    |

  4. In Figure, P Q R S is a parallelogram, P O and Q O are, respectively, ...

    Text Solution

    |

  5. The diagonals of a parallelogram A B C D intersect at OdotA line throu...

    Text Solution

    |

  6. In Figure, A B C D is a parallelogram and /D A B=60^0dot If the bisect...

    Text Solution

    |

  7. In a parallelogram A B C D , the bisector of /A also bisects B C at Xd...

    Text Solution

    |

  8. A B C D is a parallelogram. L and M are points on A B and D C respecti...

    Text Solution

    |

  9. P Q R S is a parallelogram. P X and QY are respectively, the perpendic...

    Text Solution

    |

  10. A B C D is a parallelogram. A B is produced to E so that B E=A B . Pro...

    Text Solution

    |

  11. If A B C D is a quadrilateral in which A B C D and A D=B C , prove th...

    Text Solution

    |

  12. Find the measure of all the angles of a parallelogram, if one angle is...

    Text Solution

    |

  13. The perimeter of a parallelogram is 22c mdot If the longer side measur...

    Text Solution

    |

  14. In Figure, A B C D is a parallelogram in which P is the mid-point of D...

    Text Solution

    |

  15. A B C is a triangle. D is a point on A B such that A D=1/4A B and E is...

    Text Solution

    |

  16. Show that the line segments joining the mid-points of the opposite s...

    Text Solution

    |

  17. In Figure, A N and C P are perpendicular to the diagonal B D of a para...

    Text Solution

    |

  18. Given A B C, lines are drawn through A , B and C parallel respectivel...

    Text Solution

    |

  19. In a parallelogram A B C D ,/D=115^@, determine the measure of /A and ...

    Text Solution

    |

  20. In a parallelogram A B C D , prove that sum of any two consecutive ang...

    Text Solution

    |