All the students who wish to learn all the theorems of Area must know all the properties of the area of triangle and parallelogram. We know that a quadrilateral can be divided into two triangles. In this article, we will show that “If Each Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area then Prove that the Quadrilateral is a Parallelogram”. By practicing the theorems from here you can solve any type of problem and can score good marks in the exams.
If Each Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area then Prove that the Quadrilateral is a Parallelogram
Question:
Prove that If each diagonal of a quadrilateral separates it into two triangles of equal area then show that the quadrilateral is a parallelogram?
Diagonals of quadrilatreral ABCD divides it into two triangles of equal area.
∴ ar( ∆ ABC) = ar( ∆ ADC) [Considering diagonal AC]
Now, ar( ∆ ABC) + ar( ∆ ADC) = ar (ABCD)
⇒ 2ar( ∆ ABC) = ar (ABCD)
⇒ ar( ∆ ABC) = 1/2 ar (ABCD) ….(1)
Considering diagonal BD
ar( ∆ ABD) = ar( ∆ BCD) [Diagonals of quadrilatreral ABCD divides it into two triangles of equal area.]
Now, ar( ∆ ABD) + ar( ∆ BCD) = ar (ABCD)
2ar( ∆ ABD) = ar (ABCD)
ar( ∆ ABD) = 1/2 ar (ABCD) ….(2)
From (1) and (2)
ar( ∆ ABC) = ar( ∆ ABD)
Now,
∆ABC and ∆ABD are on the same base AB and have the same areas.
∴ They lie between the same parallel lines.
∴ DC || AB
Similarly, BC || AD
In quadrilateral ABCD,
DC || AB
BC || AD
Hence ABCD is a parallelogram.
Also, See:
- Opposite Angles of a Parallelogram are Equal
- A Parallelogram, whose Diagonals are of Equal Length, is a Rectangle
- Base and Height (Altitude) in a Triangle and a Parallelogram
FAQ’s on Diagonal of a Quadrilateral Divides the Quadrilateral into Two Triangles
1. Can a quadrilateral be divided into 2 triangle?
Yes, any quadrilateral can be divided into 2 triangles. The sum of the interior angles of any triangle is 180 degrees and the sum of the angles for each quadrilateral is 360 degrees.
2. When you divide a quadrilateral diagonally you will have how many triangles?
If you divide a quadrilateral diagonally, you will have four triangles.
3. Is it true that if two diagonals of a quadrilateral divide it into 4 triangles of an equal area then the quadrilateral is a parallelogram?
The Diagonals Divide a Parallelogram Into Four Equal Areas.