Parallelogram Property Investigation: Proving AO = OC in a Quadrilateral

Q: Why do I need to check both diagonal pairs?

A parallelogram requires that both diagonals bisect each other. Finding AO = OC only proves one diagonal is bisected. You must also verify BO = OD to confirm it's truly a parallelogram.

Q: Can a quadrilateral have one bisected diagonal but not be a parallelogram?

Yes! Many quadrilaterals can have one diagonal bisected. Only when both diagonals bisect each other do you have a parallelogram.

Q: What if BO ≠ OD after finding x?

This means the quadrilateral is not a parallelogram. Even though one diagonal bisects the other, both must bisect for the parallelogram property to hold.

Parallelogram Verification with Diagonal Intersection

Look at the quadrilateral below.

AO = OC

Is it a parallelogram?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the quadrilateral below.

AO = OC

Is it a parallelogram?

Step-by-step solution

Let's pay attention to the diagonals, remember that in a parallelogram the diagonals intersect each other.

Therefore, we will find AO, OC, BO, DO and check if they are equal and intersect each other.

We refer to the figure:

$AO=OC$

$9x+1=10x$

We place like terms:

$1=10x-9x$

$1=x$

We replace:

$AO=9\times1+1=10$

$OC=10\times1=10$

Now we know that indeed $AO=OC$

Now we establish that X=1 and see if BO is equal to OD:

$BO=3x-2$

$BO=3\times1-2=$

$BO=3-2=1$

$OD=5x+4$

$OD=5\times1+4$

$OD=5+4=9$

Now we find that: $BO\ne OD$

Since the diagonals do not intersect each other, the quadrilateral is not a parallelogram.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Parallelogram diagonals must bisect each other at intersection point
Technique: Set diagonal segments equal: 9x+1 = 10x gives x = 1
Check: Both diagonal pairs must be equal: AO = OC and BO = OD ✓

Common Mistakes

Avoid these frequent errors

Stopping after finding one diagonal pair equal
Don't assume it's a parallelogram just because AO = OC! This only proves one diagonal bisects, not both. You need BOTH diagonal pairs equal: AO = OC AND BO = OD. Always check both diagonal segments before concluding.

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

Why do I need to check both diagonal pairs?

A parallelogram requires that both diagonals bisect each other. Finding AO = OC only proves one diagonal is bisected. You must also verify BO = OD to confirm it's truly a parallelogram.

What if I get different values for x from each diagonal?

If the diagonals give different x values, the quadrilateral is definitely not a parallelogram. The intersection point must work for both diagonals simultaneously.

Can a quadrilateral have one bisected diagonal but not be a parallelogram?

Yes! Many quadrilaterals can have one diagonal bisected. Only when both diagonals bisect each other do you have a parallelogram.

How do I set up the equations correctly?

Identify the intersection point O, then set equal segments:

AO = OC (one diagonal)
BO = OD (other diagonal)

Use the same x value for both equations.

What if BO ≠ OD after finding x?

This means the quadrilateral is not a parallelogram. Even though one diagonal bisects the other, both must bisect for the parallelogram property to hold.

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