Home
Class 9
MATHS
Show that the diagonals of a parallel...

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

Text Solution

Verified by Experts

GIVEN A ||gm ABCD. Its diagonals AC and BD intersect at O.
TO PROVE `ar(triangleOAB)=ar(triangleOBC)`
`=ar(triangleOCD)=ar(triangleOAD)`

PROOF Since the diagonals of a ||gm bisect each other, we have
`OA=OC and OB=OD`.
Also, a median of a triangle divides it into two `triangle` of equal area.
Now, in `triangle`ABC, BO is the median.
`therefore ar(triangleOAB)=ar(triangleOBC)." "...(i)`
In `triangleABD,AO` is the median.
`therefore ar(triangleOAB)=ar(triangleOAD)." "...(ii)`
In `triangleACD,DO` is the median.
`therefore ar(triangleOAD)=ar(triangleOCD)." "...(iii)`
From (i), (ii) and (iii), we get
`therefore ar(triangleOAB)=ar(triangleOBC)." "...(i)`
Promotional Banner

Topper's Solved these Questions

  • AREAS OF PARALLELOGRAMS AND TRIANGLES

    RS AGGARWAL|Exercise Exercise 11|38 Videos

  • AREAS OF PARALLELOGRAMS AND TRIANGLES

    RS AGGARWAL|Exercise MULTIPLE-CHOICE QUESTIONS (MCQ)|32 Videos

  • AREAS OF TRIANGLES AND QUADRILATERALS

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

Similar Questions

Explore conceptually related problems

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

A diagonal of a parallelogram divides it into two congruent triangles.

A diagonal of parallelogram divides it into two congruent triangles.

A diagonal of a parallelogram divides it into two triangles of equal area. GIVEN : A parallelogram A B C D in which B D is one of the diagonals. TO PROVE : ar ( A B D)=a r( C D B)

The diagonals of a parallelograms are equal .

Show that each diagonal of a parallelogram divide it into two congruent triangles. The following are the steps involved in showing the above result. Arrange them in sequential order. A) In triangleABC and triangleCDA , AB=DC and BC=AD (therefore opposite angles of parallelogram) AC=AC (common side). B) Let ABCD be a parallelogram. Join AC. C) By SSS congruence property, triangleABC ~=triangleCDA . D) Similarly, BD divides the triangle into two congruent triangles.

Show that a median of a triangle divides it into two triangles of equal areas.

Show that a median of a triangle divides it into two triangles of equal area.

Asserion (A) : The diagonals of a ||gm divide it into four triangle of equal area. Reason (R ) : A diagonal of a ||gm divides it into two triangle of equal area.