Home
Class 9
MATHS
If each diagonals of a quadrilateral ...

If each diagonals of a quadrilateral separates it into two triangles of equal area then show that the quadrilateral is a parallelogram.

Text Solution

Verified by Experts

Let AC and BD are two diagonals of quadrilateral ABCD.
Given that, the diagonal AC divides `square ABCD` into two triangles of equal areas.
`:.` area of `Delta ABC = " area of " DeltaACD`
`rArr` area of `Delta ABC = (1)/(2) xx " Area of " square ABCD`...(1)
Similarly, the diagonal BD of `square ABCD` divides it into two triangles of equal areas.
`:.` area of `Delta ABD = " area of " Delta BCD`
`rArr` area of `Delta ABD = (1)/(2) xx " area of " square ABCD`...(2)
From (1) and (2), we get
area of `Delta ABC = " area of " Delta ABD`
But these are on the same base AB. (by theorem 5)
`:.` Their corresponding height will be same.
`rArr AB and DC` are parallel.
Similarly, we can prove that `AD ||BC`
`:. square ABCD` is a parallelogram
Promotional Banner

Topper's Solved these Questions

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    NAGEEN PRAKASHAN|Exercise Problems From NCERT/exemplar|12 Videos

  • AREA OF PARALLELOGRAMS AND TRIANGLES

    NAGEEN PRAKASHAN|Exercise Exercise|34 Videos

  • CIRCLE

    NAGEEN PRAKASHAN|Exercise Revision Exercise (long Answer Questions )|5 Videos

Similar Questions

Explore conceptually related problems

If each diagonal of a quadrilateral separates it into two triangles of equal area then show that the quadrilateral is a parallelogram. GIVEN : A quadrilateral A B C D such that its diagonals A C and B D are such that a r( A B D)=a r( C D B ) and a r( A B C)=a r( A C D)dot TO PROVE: Quadrilateral A B C D is a parallelogram.

A diagonal of a parallelogram divides it into two triangles of equal area.

If the diagonals AC,BD of a quadrilateral ABCD, intersect at O, and separate the quadrilateral into four triangles of equal area, show that quadrilateral ABCD is a parallelogram.

If the diagonals A C ,B D of a quadrilateral A B C D , intersect at O , and seqarate the quadrilateral into four triangles of equal area, show that quadrilateral A B C D is a parallelogram. GIVEN : A quadrilateral A B C D such that its diagonals A C and B D intersect at O and separate it into four parts such that a r( A O B)=a r( B O C)=a r( C O D)=a r( A O D) TO PROVE : Quadrilateral A B C D is a parallelogram.

Show that the diagonals of a parallelogram divide it into four triangles of equal area.

Show that the diagonals of a parallelogram divide it into four triangles of equal area.

Show that the diagonals of a parallelogram divide it into four triangles of equal area.

If the diagonals of a quadrilateral bisect each other,then the quadrilateral is a parallelogram.

If the opposite sides of a quadrilateral are equal, prove that the quadrilateral is a parallelogram.