Calculate Parallelogram Area: Using 6-Unit Height and 9-Unit Base with Perpendicular Lines

Q: How do I know which measurement is the height?

The height is always the perpendicular distance between parallel sides. Look for the ⊥ symbol or words like 'perpendicular' - that's your height! Here, BF ⊥ DE means BF = 6 is the height.

Q: Why does angle ACB = angle EBC prove AC || BE?

These are alternate angles! When a line crosses two other lines and creates equal alternate angles, those two lines must be parallel. It's a key geometry rule.

Q: What if I accidentally use the wrong formula?

For parallelograms, always use Area = base × height, not $ \frac{1}{2} \times base \times height $ (that's for triangles!). Double-check which shape you're working with first.

Parallelogram Area with Perpendicular Height Lines

ABCD is a parallelogram.

Angle ACB is equal to angle EBC.

BF = 6

CE = 9

BF is perpendicular to DE.

Calculate the area of the parallelogram.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of parallelogram ABCD

00:04 Alternate angles are equal between parallel lines

00:15 A line parallel to another line is also parallel to its extension

00:22 Quadrilateral ABCE is a parallelogram

00:27 Opposite sides are equal in a parallelogram

00:31 We'll use the formula to calculate the area of a parallelogram

00:35 Side(AB) multiplied by height (BF)

00:39 We'll substitute appropriate values and solve to find the area

00:42 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

ABCD is a parallelogram.

Angle ACB is equal to angle EBC.

BF = 6

CE = 9

BF is perpendicular to DE.

Calculate the area of the parallelogram.

Step-by-step solution

Given that angle ACB is equal to angle CBE, it follows that AC is parallel to BE

since alternate angles between parallel lines are equal.

As we know that ABCD is a parallelogram, AB is parallel to DC and therefore AB is also parallel to CE since it is a line that continues DC.

Given that AC is parallel to BE and, in addition, AB is parallel to CE, it can be argued that ABCE is a parallelogram and, therefore, each pair of opposite sides in a parallelogram are parallel and equal.

From this it is concluded that AB=CE=9

Now we calculate the area of the parallelogram ABCD according to the data.

$S_{ABCD}=AB\times BF$

We replace the data accordingly:

$S_{ABCD}=9\times6=54$

Final Answer

54 cm²

Key Points to Remember

Essential concepts to master this topic

Parallel Lines: Equal alternate angles prove lines are parallel
Technique: ABCE forms parallelogram, so AB = CE = 9
Check: Area = base × height = 9 × 6 = 54 cm² ✓

Common Mistakes

Avoid these frequent errors

Using slanted side length instead of perpendicular height
Don't measure diagonal BC or slanted sides = wrong area calculation! Slanted measurements don't give true height needed for area formula. Always use perpendicular distance BF = 6 as the height.

Practice Quiz

Test your knowledge with interactive questions

A parallelogram has a length equal to 6 cm and a height equal to 4.5 cm.

Calculate the area of the parallelogram.

FAQ

Everything you need to know about this question

How do I know which measurement is the height?

The height is always the perpendicular distance between parallel sides. Look for the ⊥ symbol or words like 'perpendicular' - that's your height! Here, BF ⊥ DE means BF = 6 is the height.

Why does angle ACB = angle EBC prove AC || BE?

These are alternate angles! When a line crosses two other lines and creates equal alternate angles, those two lines must be parallel. It's a key geometry rule.

How did we figure out that AB = 9?

Since ABCE is a parallelogram (both pairs of opposite sides are parallel), opposite sides are equal. So AB = CE = 9. This is a fundamental property of all parallelograms!

Can I use a different base and height for the same area?

Yes! You could use base AD with height from B to AD, or base BC with its perpendicular height. All methods give the same area when done correctly.

What if I accidentally use the wrong formula?

For parallelograms, always use Area = base × height, not $\frac{1}{2} \times base \times height$ (that's for triangles!). Double-check which shape you're working with first.

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