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If the diagonals A C ,\ B D of a qua...

If the diagonals `A C ,\ B D` of a quadrilateral `A B C D ,` intersect at `O ,` and separate the quadrilateral into four triangles of equal area, show that quadrilateral `A B C D` is a parallelogram.

Text Solution

Verified by Experts

Given
`ar(AOD)=ar(BOC)`
Add area`(AOB)` to both sides
`ar(AOD)+ar(AOB)=ar(BOC)+ ar(AOB)`
`=>ar(ABD)=ar(ABC)`
`/_\ABD and /_\ABC`,lie on the same base `AB` and have equal area.
`:. AD|\ | BC`
Similarly
`DC|\ |AB`
Opposite side of quadrilateral `ABCD` are parallel
`:.ABCD` is a `|\ |gm`
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RD SHARMA-AREA OF PARALLELOGRAMS AND TRIANGLES-All Questions
  1. Show that the diagonals of a parallelogram divide it into four trian...

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  2. The diagonals of quadrilateral A B C D ,AC and B D intersect in O...

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  3. If the diagonals A C ,\ B D of a quadrilateral A B C D , intersect...

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  4. If each diagonals of a quadrilateral separates it into two triangles...

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  5. Show that the area of a rhombus is half the product of the lengths o...

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  6. The side AB of a parallelogram ABCD is produced to any point P. A line...

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  7. A villager Itwari has a plot of land of the shape of a quadrilateral...

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  8. A B C D is a trapezium with A B|\|D C . A line parallel to A C int...

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  9. Diagonals A C\ a n d\ B D of a quadrilateral A B C D intersect a...

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  10. In Figure, A P B Q\ C Rdot Prove that a r\ ( A Q C)=\ a r\ (P B R)

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  11. In Fig.9.29, a r\ (B D P)\ =\ a r\ (A R C)and a r\ (B D P)\ =\ a r\ (A...

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  12. Diagonals A C and B D of a quadrilateral A B C D intersect at O in suc...

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  13. In Fig. 9.25, diagonals AC and BD of quadrilateral ABCD intersect at O...

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  14. A point O inside a rectangle A B C D is joined to the vertices. Prov...

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  15. Show that the area of a rhombus is half the product of the lengths ...

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  16. A B C D IS A PARALLELOGRAM AND O is any point in its interior. Prov...

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  17. A quadrilateral A B C D is such that diagonal B D divides its area in ...

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  18. Parallelogram A B C D and rectangle A B E F have the same base A B and...

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  19. O is any point on the diagonal B D of the parallelogram A B C Ddot Pro...

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  20. Triangles A B C and +DBC are on the same base B C with A, D on opposit...

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