Study Guides/Maths/Are Diagonals of a Parallelogram Equal?
Study Guide · Maths

Are the Diagonals of a Parallelogram Equal?

One of the most frequently asked questions in Class 8 and 9 Geometry is whether the diagonals of a parallelogram are equal in length.

The short answer is: No, the diagonals of a general parallelogram are NOT equal.

Question (Click to Flip)

Do the diagonals of a parallelogram divide it into equal triangles?

Answer

Yes! A single diagonal divides a parallelogram into two congruent triangles. If you draw both diagonals, they divide the parallelogram into four triangles of equal area (though they are not all congruent to each other).

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Key Facts

If you are given a parallelogram and told that its diagonals are equal, you have just proved that the parallelogram is actually a rectangle. This is a standard theorem proof in Class 9 Maths.

Diagonals of a Standard Parallelogram

In a standard parallelogram (where all angles are not 90° and all sides are not equal):

  1. They are NOT equal: One diagonal is longer than the other.
  2. They DO bisect each other: They cut each other exactly in half. If the diagonals are AC and BD intersecting at O, then AO = OC and BO = OD.
  3. They are NOT perpendicular: They do not cross at 90° angles.

When ARE the Diagonals Equal?

The diagonals of a parallelogram are equal ONLY if the parallelogram is a Rectangle or a Square.

Because rectangles and squares have 90° angles, the distance between the opposite corners becomes exactly the same.

  • Rectangle: Diagonals are Equal & Bisect each other.
  • Square: Diagonals are Equal, Bisect each other, AND are Perpendicular (90°).

What about a Rhombus?

A rhombus is a special parallelogram where all 4 sides are equal.

  • Are its diagonals equal? No.
  • Do they bisect each other? Yes.
  • Are they perpendicular? Yes (They cross exactly at 90°).

So, a rhombus has perpendicular diagonals, but they are not equal in length (unless the rhombus is a square).

Summary Table of Quadrilateral Diagonals

QuadrilateralAre Diagonals Equal?Do they Bisect?Are they Perpendicular?
Parallelogram❌ No✅ Yes❌ No
Rectangle✅ Yes✅ Yes❌ No
Rhombus❌ No✅ Yes✅ Yes
Square✅ Yes✅ Yes✅ Yes

Questions and Answers

Do the diagonals of a parallelogram divide it into equal triangles?+

Yes! A single diagonal divides a parallelogram into **two congruent triangles**. If you draw both diagonals, they divide the parallelogram into **four triangles of equal area** (though they are not all congruent to each other).

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