Parallelogram Property Test: Analyzing Segments of 2 and 5 Units in ABCD

Q: What does it mean for diagonals to bisect each other?

When diagonals bisect each other, they cut each other exactly in half at their intersection point. This means $AF = FD$ and $BF = FC$.

Q: Are there other ways to prove a quadrilateral is a parallelogram?

Yes! You can also prove it by showing: Opposite sides are parallelOpposite sides are equalOpposite angles are equalOne pair of opposite sides is both parallel and equal

Q: What if only one diagonal is bisected by the intersection point?

Then it's not a parallelogram! Both diagonals must bisect each other for the quadrilateral to be a parallelogram. Having just one bisected diagonal isn't enough.

Q: Does the length of the diagonal segments matter?

No, the actual lengths don't matter! What matters is that each diagonal is split into two equal parts. Here we have 2 = 2 and 5 = 5, but 1 = 1 and 10 = 10 would work just as well.

Q: How can I remember which segments to compare?

Think of it as same diagonal, same comparison! For diagonal AC, compare AF with FC. For diagonal BD, compare BF with FD. Never mix segments from different diagonals.

Parallelogram Properties with Diagonal Bisection

Look at the quadrilateral ABCD shown below.

AF = 2 and FD = 2.

BF = 5 and FC = 5.

Is this quadrilateral a parallelogram?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the square a parallelogram?

00:03 The length of the diagonals according to the given data

00:08 A quadrilateral whose diagonals intersect each other is a parallelogram

00:12 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the quadrilateral ABCD shown below.

AF = 2 and FD = 2.

BF = 5 and FC = 5.

Is this quadrilateral a parallelogram?

Step-by-step solution

To determine if quadrilateral $ABCD$ is a parallelogram, we need to verify if its diagonals $AC$ and $BD$ bisect each other. This can be confirmed if their respective segments around intersection point $F$ are equal.

Given the information, we have:

$AF = 2$ and $FD = 2$ , indicating diagonal $AD$ is bisected by $F$ .
$BF = 5$ and $FC = 5$ , indicating diagonal $BC$ is bisected by $F$ .

Since both conditions are satisfied, the diagonals indeed bisect each other, fulfilling the criteria for quadrilateral $ABCD$ to be a parallelogram.

Therefore, the answer to the question is Yes, quadrilateral $ABCD$ is a parallelogram.

Final Answer

Yes.

Key Points to Remember

Essential concepts to master this topic

Rule: Parallelogram diagonals bisect each other at their intersection
Technique: Check if AF = FD and BF = FC, here 2 = 2 and 5 = 5
Check: Both diagonal segments are equal around point F, confirming parallelogram ✓

Common Mistakes

Avoid these frequent errors

Confusing which segments to compare
Don't compare AF with BF or FC with FD = wrong diagonal analysis! This mixes segments from different diagonals and gives incorrect conclusions. Always compare segments from the same diagonal: AF with FD, and BF with FC.

Practice Quiz

Test your knowledge with interactive questions

Shown below is the quadrilateral ABCD.

AB = 15 and CD = 13.

BD = 6 and AC = 4

Is it possible to conclude that this quadrilateral is a parallelogram?

FAQ

Everything you need to know about this question

What does it mean for diagonals to bisect each other?

When diagonals bisect each other, they cut each other exactly in half at their intersection point. This means $AF = FD$ and $BF = FC$ .

Are there other ways to prove a quadrilateral is a parallelogram?

Yes! You can also prove it by showing:

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

What if only one diagonal is bisected by the intersection point?

Then it's not a parallelogram! Both diagonals must bisect each other for the quadrilateral to be a parallelogram. Having just one bisected diagonal isn't enough.

Does the length of the diagonal segments matter?

No, the actual lengths don't matter! What matters is that each diagonal is split into two equal parts. Here we have 2 = 2 and 5 = 5, but 1 = 1 and 10 = 10 would work just as well.

How can I remember which segments to compare?

Think of it as same diagonal, same comparison! For diagonal AC, compare AF with FC. For diagonal BD, compare BF with FD. Never mix segments from different diagonals.

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