Parallelogram Investigation: Analyzing Segments AF=4, FD=6, BF=2, FC=3

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

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

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

Yes! You can also check if opposite sides are parallel and equal, or if opposite angles are equal. But the diagonal test is often the quickest method when given intersection segments.

Q: Why can't this be a parallelogram if AF ≠ FD?

In a parallelogram, the diagonals must bisect each other. Since $ AF = 4 $ and $ FD = 6 $, diagonal AD is not bisected at F, violating this essential property.

Q: Could this quadrilateral be some other special shape?

Possibly! It could be a trapezoid or just an irregular quadrilateral. The given information only tells us it's not a parallelogram, rectangle, rhombus, or square.

Q: Do I need to check both diagonals or just one?

Check both diagonals! For a parallelogram, both diagonals must bisect each other. If even one diagonal fails the test, it's not a parallelogram.

Parallelogram Properties with Diagonal Intersection

Below is the quadrilateral ABCD.

AF = 4 and FD = 6.

BF = 2 and FC = 3.

Is the quadrilateral a parallelogram?

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Step-by-step video solution

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00:10 Is the shape a parallelogram? Let's find out.

00:15 In a parallelogram, the diagonals should meet at a point. Let's check for this.

00:23 The diagonals do not meet at a point. So, it's not a parallelogram.

00:28 And that's how we solve this problem. Well done!

Step-by-step written solution

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Understand the problem

Below is the quadrilateral ABCD.

AF = 4 and FD = 6.

BF = 2 and FC = 3.

Is the quadrilateral a parallelogram?

Step-by-step solution

To determine whether quadrilateral ABCD is a parallelogram, we examine the segments of its diagonals.

Measure segments $AF$ and $FD$ on diagonal $AD$ .
Measure segments $BF$ and $FC$ on diagonal $BC$ .

We have:

$AF = 4$ and $FD = 6$ .
$BF = 2$ and $FC = 3$ .

A necessary condition for ABCD to be a parallelogram is that its diagonals bisect each other:

This requires $AF = FD$ and $BF = FC$ .

Here, $AF$ is not equal to $FD$ and $BF$ is not equal to $FC$ , which means the diagonals do not bisect each other.

Therefore, quadrilateral ABCD is not a parallelogram.

The correct answer is No.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Parallelogram diagonals must bisect each other at intersection point
Technique: Check if AF = FD and BF = FC; here 4 ≠ 6 and 2 ≠ 3
Check: If any diagonal segments are unequal, then not a parallelogram ✓

Common Mistakes

Avoid these frequent errors

Assuming any quadrilateral with intersecting diagonals is a parallelogram
Don't just look at whether diagonals cross = wrong conclusion! Diagonals can intersect without bisecting each other. Always verify that each diagonal is split into two equal segments at the intersection point.

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. This means $AF = FD$ and $BF = FC$ at the intersection point F.

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

Yes! You can also check if opposite sides are parallel and equal, or if opposite angles are equal. But the diagonal test is often the quickest method when given intersection segments.

Why can't this be a parallelogram if AF ≠ FD?

In a parallelogram, the diagonals must bisect each other. Since $AF = 4$ and $FD = 6$ , diagonal AD is not bisected at F, violating this essential property.

Could this quadrilateral be some other special shape?

Possibly! It could be a trapezoid or just an irregular quadrilateral. The given information only tells us it's not a parallelogram, rectangle, rhombus, or square.

Do I need to check both diagonals or just one?

Check both diagonals! For a parallelogram, both diagonals must bisect each other. If even one diagonal fails the test, it's not a parallelogram.

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