Parallelogram Analysis: Investigating a Quadrilateral with 8 and 10 Unit Diagonals

Q: What makes this quadrilateral a parallelogram?

The diagonals bisect each other! Since AO = OC = 8 and DO = OB = 10, point O is the midpoint of both diagonals. This is a key property that confirms it's a parallelogram.

Q: Do the diagonals need to be the same length?

No! Parallelogram diagonals can have different lengths. Here, diagonal AC = 16 and diagonal BD = 20. What matters is that they bisect each other, not that they're equal.

Q: What if only one diagonal was bisected?

Then it wouldn't be a parallelogram! Both diagonals must bisect each other for the quadrilateral to be a parallelogram. This is a necessary and sufficient condition.

Parallelogram Properties with Diagonal Intersection

Below is a quadrilateral:

Is it possible that it is a parallelogram?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is a quadrilateral:

Is it possible that it is a parallelogram?

Step-by-step solution

According to the properties of the parallelogram: the diagonals intersect each other.

From the data in the drawing, it follows that diagonal AC and diagonal BD are divided into two equal parts, that is, the diagonals intersect each other:

$AO=OC=8$

$DO=OB=10$

Therefore, the quadrilateral is actually a parallelogram.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Property: Parallelogram diagonals bisect each other at intersection point
Technique: Check if AO = OC = 8 and DO = OB = 10
Check: Both diagonals split into equal halves at point O ✓

Common Mistakes

Avoid these frequent errors

Only checking if diagonals are equal in length
Don't assume it's a parallelogram just because diagonals AC = BD = 20! Equal diagonal lengths don't guarantee a parallelogram. Always check that diagonals bisect each other: AO = OC and DO = OB.

Practice Quiz

Test your knowledge with interactive questions

It is possible to draw a quadrilateral that is not a rectangle, with the sum of its two adjacent angles equaling 180?

FAQ

Everything you need to know about this question

What makes this quadrilateral a parallelogram?

The diagonals bisect each other! Since AO = OC = 8 and DO = OB = 10, point O is the midpoint of both diagonals. This is a key property that confirms it's a parallelogram.

Do the diagonals need to be the same length?

No! Parallelogram diagonals can have different lengths. Here, diagonal AC = 16 and diagonal BD = 20. What matters is that they bisect each other, not that they're equal.

How do I know the diagonals bisect each other?

Look at the intersection point O. Measure from each vertex to O:

AO = 8, OC = 8 (diagonal AC is bisected)
DO = 10, OB = 10 (diagonal BD is bisected)

What if only one diagonal was bisected?

Then it wouldn't be a parallelogram! Both diagonals must bisect each other for the quadrilateral to be a parallelogram. This is a necessary and sufficient condition.

Are there other ways to prove it's a parallelogram?

Yes! You could also check if:

Opposite sides are parallel and equal
Opposite angles are equal
One pair of opposite sides is both parallel and equal

But the diagonal bisection method shown here is often the quickest!

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