Home
Class 9
MATHS
In the adjoining figure, ABCD is a paral...

In the adjoining figure, ABCD is a parallelogram in which X and Y are the midpoints of AB and DC respectively. If AY and DX intersect in P while CX and BY intersect in Q, show that

(i) quad. AXCY is a parallelogram,
(ii) quad. XBYD is a parallelogram,
(iii) quad. PXQY is a parallelogram.

Text Solution

Verified by Experts

`AB"||"DC rArr AX "||"YC.`
Also, `AB = DC rArr (1)/(2)AB =(1)/(2)DC rArr AX=YC.`
`therefore "AX||YC" and AX=YC rArr AXCY " is a ||gm". `
`therefore` AY||XC. In particualr, PY||XQ.
Again, AB||DC `rArr` XB||DY.
Also, `AB=DC rArr (1)/(2)AB =(1)/(2) DC rArr XB=DY.`
` therefore "XB||DY " and XB=DY rArr XBYD " is a ||gm." `
`therefore` XD||BY. In particular, XP||QY.
Thus, PY||XQ and XP||QY.
`therefore ` PXQY is a parallelogram.
Promotional Banner

Topper's Solved these Questions

  • QUADRILATERALS

    RS AGGARWAL|Exercise Exercise 10 A|10 Videos

  • QUADRILATERALS

    RS AGGARWAL|Exercise EXERCISE 10B|27 Videos

  • QUADRILATERALS

    RS AGGARWAL|Exercise Matching of Columns:|2 Videos

  • PROBABILITY

    RS AGGARWAL|Exercise Multiple Choice Questions (Mcq)|16 Videos

  • TRIANGLES

    RS AGGARWAL|Exercise MULTIPLE-CHOICE QUESTIONS (MCQ)|10 Videos

Similar Questions

Explore conceptually related problems

ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD respectively (see figure). If AQ intersects DP at S and BQ intersects CP at R, show that: (i) APCQ is a parallelogram.

In the given figure ABCD is a parallelogram in which AB||CD and E and F are the mid points of AB and BC respectively. What is the area of angleBEF ?

ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD respectively (see figure). If AQ intersects DP at S and BQ intersects CP at R, show that: DPBQ is a parallelogram

ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD respectively (see figure). If AQ intersects DP at S and BQ intersects CP at R, show that: PSQR is a parallelogram.

ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD (see Fig. 8.18). If AQ intersects DP at S and BQ intersects CP at R, show that: (i) APCQ is a parallelogram. (ii) DPBQ is a parallelogram. (iii) PSQR is a parallelogram

ABCD is a parallelogram,E and F are the mid-points of AB and CD respectively.GH is any line intersecting AD, EF and BC at G,P and H respectively.Prove that GP=PH

ABCD is a parallelogram. P and R are the midpoints of DC and BC respectively. The line PR intersects the diagonal AC at Q . The distance CQ will be equal to

In Figure, A B C D is a parallelogram and X ,Y are the mid=points of sides A B and D C respectively. Show that A X C Y is a parallelogram. Figure

ABCD is a parallelogram. E and F are mid-point of side AB and CD respectively. AF and CE intersect diagonal BD at P & Q respectively. If BD = 15 cm, find PQ.

In Figure, X ,\ Y are the mid-points of opposite sides A B\ a n d\ D C of a parallelogram A B C DdotA Y\ a n d\ D X are joined intersecting in P ; C X\ a n d\ B Y are joined intersecting in Q . Show that A X C Y is a parallelogram (ii) D X B Y\ is a parallelogram quadrilateral P X Q Y is a parallelogram

RS AGGARWAL-QUADRILATERALS-EXAMPLE
  1. If a diagonal of a parallelogram bisects one of the angles of the para...

    Text Solution

    |

  2. Let triangle ABC and triangle DEF be two triangles given in such a way...

    Text Solution

    |

  3. In the adjoining figure, ABCD is a parallelogram in which X and Y are ...

    Text Solution

    |

  4. In the adjoining figure, ABCD is a parallelogram and E is the midpoint...

    Text Solution

    |

  5. In the adjoining figure, AB = AC: CP||BA and AP is the bisector of ang...

    Text Solution

    |

  6. In the adjoining figure, ABCD is a parallelogram and the bisector of...

    Text Solution

    |

  7. In the adjoining figure, ABCD is a parallelogram, E is midpoint of AB ...

    Text Solution

    |

  8. ABCD is a trapezium in which AB||CD and AD = BC. Show that (i) angl...

    Text Solution

    |

  9. ABCD is a parallelogram in which E and F are the midpoints of the side...

    Text Solution

    |

  10. Let ABCD be a trapezium in which AB||DC and let E be the midpoint of A...

    Text Solution

    |

  11. In the adjoining figure, ABCD is a ||gm in which P is the midpoint of...

    Text Solution

    |

  12. In the adjoining figure, AD is a median of triangle ABC and E is the m...

    Text Solution

    |

  13. In the adjoining figure, ABCD is a trapezium in which AB||DC and AD = ...

    Text Solution

    |

  14. In the adjoining figure, ABCD and PQRC are rectangles, where Q is the ...

    Text Solution

    |

  15. In the given figure, D, E and F are the midpoints of the sides BC, CA ...

    Text Solution

    |

  16. In the adjoining figure, two points A and B lie on the same side of a ...

    Text Solution

    |

  17. In the adjoining figure, points M and N divide the side AB of triangl...

    Text Solution

    |

  18. E and F are respectively the midpoints of nonparallel sides AD and BC ...

    Text Solution

    |

  19. Prove that any line segment drawn from the vertex of a triangle to the...

    Text Solution

    |

  20. In the adjoining figure, the side AC of triangleABC is produced to E s...

    Text Solution

    |