Diagonal Properties of Parallelogram
The diagonals of a parallelogram have a key property: they bisect each other. This means each diagonal cuts the other into two equal halves at their point of intersection.
This property is both a theorem and a test — it can be used to prove a quadrilateral is a parallelogram, and it follows from the definition. The converse is also true: if the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
What is Diagonal Properties of Parallelogram?
Theorem: The diagonals of a parallelogram bisect each other.
Converse: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
In parallelogram ABCD with diagonals AC and BD intersecting at O:
AO = OC and BO = OD
Diagonal Properties of Parallelogram Formula
Special cases:
- Rectangle: Diagonals bisect each other AND are equal (AC = BD).
- Rhombus: Diagonals bisect each other AND are perpendicular (AC ⊥ BD).
- Square: Diagonals bisect each other, are equal, AND are perpendicular.
Derivation and Proof
Proof that diagonals of a parallelogram bisect each other:
- Let ABCD be a parallelogram with diagonals AC and BD intersecting at O.
- In △AOB and △COD:
- AB ∥ CD (opposite sides of parallelogram) → ∠OAB = ∠OCD (alternate interior angles with transversal AC)
- ∠OBA = ∠ODC (alternate interior angles with transversal BD)
- AB = CD (opposite sides of parallelogram)
- By ASA congruence: △AOB ≅ △COD.
- Therefore: AO = OC and BO = OD (CPCT).
- Hence the diagonals bisect each other. ∎
Solved Examples
Example 1: Example 1: Finding half-diagonal lengths
Problem: ABCD is a parallelogram. AC = 18 cm and BD = 12 cm. Find AO, OC, BO, OD.
Solution:
- AO = OC = AC/2 = 18/2 = 9 cm
- BO = OD = BD/2 = 12/2 = 6 cm
Answer: AO = OC = 9 cm, BO = OD = 6 cm.
Example 2: Example 2: Using algebra
Problem: In parallelogram PQRS, diagonals meet at O. If PO = 3x + 1 and OR = 7x − 11, find x and PR.
Solution:
- Diagonals bisect: PO = OR
- 3x + 1 = 7x − 11
- 12 = 4x → x = 3
- PO = 3(3) + 1 = 10. PR = 2 × 10 = 20.
Answer: x = 3, PR = 20.
Example 3: Example 3: Proving it is a parallelogram
Problem: ABCD has diagonals AC and BD meeting at O. AO = 7, OC = 7, BO = 5, OD = 5. Is ABCD a parallelogram?
Solution:
- AO = OC (diagonals bisect) ✓
- BO = OD (diagonals bisect) ✓
- By the converse theorem, ABCD is a parallelogram.
Answer: Yes, ABCD is a parallelogram.
Example 4: Example 4: Rectangle diagonal property
Problem: ABCD is a rectangle with AC = 26 cm. Find AO.
Solution:
- In a rectangle, diagonals are equal and bisect each other.
- AO = AC/2 = 26/2 = 13 cm
- Also, BD = AC = 26 cm, so BO = 13 cm.
Answer: AO = 13 cm.
Example 5: Example 5: Rhombus diagonal property
Problem: ABCD is a rhombus with diagonals 10 cm and 24 cm. Find the side.
Solution:
Diagonals bisect each other at right angles in a rhombus.
- Half-diagonals: 5 cm and 12 cm
- Side = √(5² + 12²) = √(25 + 144) = √169 = 13 cm
Answer: Side = 13 cm.
Example 6: Example 6: Finding coordinates of intersection
Problem: Parallelogram ABCD has A(1, 2), B(4, 6), C(7, 4), D(4, 0). Find the point of intersection of diagonals.
Solution:
Midpoint of AC = ((1+7)/2, (2+4)/2) = (4, 3).
Midpoint of BD = ((4+4)/2, (6+0)/2) = (4, 3).
Both midpoints are the same, confirming the diagonals bisect each other.
Answer: Diagonals intersect at (4, 3).
Example 7: Example 7: Area using diagonals (rhombus)
Problem: A rhombus has diagonals 16 cm and 30 cm. Find the area.
Solution:
- Area = ½ × d₁ × d₂ = ½ × 16 × 30 = 240 cm²
Answer: Area = 240 cm².
Example 8: Example 8: Distinguishing parallelogram types
Problem: A parallelogram has diagonals that bisect each other at right angles but are not equal. What type is it?
Solution:
- Diagonals bisect each other → parallelogram ✓
- Diagonals are perpendicular → rhombus ✓
- Diagonals are NOT equal → not a square
Answer: It is a rhombus.
Real-World Applications
Applications:
- Construction: Verifying parallelogram shapes in building layouts.
- Engineering: Linkage mechanisms that use parallelogram properties.
- Coordinate geometry: Finding missing vertices using the midpoint/diagonal property.
- Proofs: Many geometry proofs use the diagonal bisection property as a key step.
Key Points to Remember
- In a parallelogram, diagonals bisect each other.
- The converse is also true: if diagonals bisect, it is a parallelogram.
- Rectangle: diagonals bisect AND are equal.
- Rhombus: diagonals bisect AND are perpendicular.
- Square: diagonals bisect, are equal, AND are perpendicular.
- The point of intersection of diagonals is the midpoint of each diagonal.
- In coordinate geometry, verify by checking midpoints of both diagonals are equal.
- Side of rhombus = √(half-diagonal₁² + half-diagonal₂²) using Pythagoras.
Practice Problems
- ABCD is a parallelogram with AC = 20 cm, BD = 14 cm. Find AO and BO.
- In parallelogram KLMN, KO = 2x+3 and OM = 5x−6. Find x and KM.
- Prove that the diagonals of a rectangle are equal.
- A rhombus has diagonals 12 and 16 cm. Find its side and area.
- ABCD has A(0,0), B(6,0), C(8,4), D(2,4). Show it is a parallelogram using the diagonal property.
- Distinguish between rectangle, rhombus, and square based on diagonal properties.
Frequently Asked Questions
Q1. Do the diagonals of a parallelogram bisect each other?
Yes. This is a proven theorem. Each diagonal is cut into two equal halves at the point of intersection.
Q2. Are the diagonals of a parallelogram equal?
Not necessarily. They are equal only in a rectangle (or square). In a general parallelogram, they can be different lengths.
Q3. Are the diagonals perpendicular in a parallelogram?
Not necessarily. They are perpendicular only in a rhombus (or square).
Q4. How do you prove a quadrilateral is a parallelogram using diagonals?
Show that the diagonals bisect each other (their midpoints coincide). This is the converse of the diagonal property.
Q5. Can you find the area of a parallelogram using diagonals?
Only for a rhombus: Area = ½ × d₁ × d₂. For a general parallelogram, you need base × height.
Q6. What is the diagonal property that distinguishes a square from a rhombus?
Both have perpendicular diagonals. But a square has equal diagonals, while a rhombus generally does not.
