Parallelogram Area Calculation: Using Height CE and Segments 5, 7, and 2

Q: Why can't I just use CB = 5 as the height?

CB is the slanted side of the parallelogram, not the height! The height must be perpendicular to the base. CE is the perpendicular distance from C to line AB.

Q: How do I know triangle EBC is a right triangle?

CE is described as the height of the parallelogram, which means it's perpendicular to the base AB. This creates a right angle at E, making triangle EBC a right triangle.

Q: Why do I get a decimal answer instead of a whole number?

Because the height $ \sqrt{21} $ is an irrational number (approximately 4.58). When multiplied by 9, you get 41.24. This is perfectly normal in geometry!

Q: How can I check if 41.24 is correct?

Verify that $ \sqrt{21} \times 9 = 41.24 $ by calculating: $ \sqrt{21} \approx 4.583 $, so $ 4.583 \times 9 \approx 41.24 $ ✓

Q: Could I solve this problem differently?

You could use coordinate geometry or trigonometry, but the Pythagorean theorem method is the most straightforward since you're given a right triangle with two known sides.

Parallelogram Area with Pythagorean Theorem

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

What is 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 the parallelogram

00:03 We'll use the Pythagorean theorem in triangle EBC

00:10 We'll substitute appropriate values and solve for EC

00:17 We'll isolate EC

00:27 This is the length of EC, which is the height of the parallelogram

00:33 To find the area, multiply the height(EC) by the side(AB)

00:40 The entire side equals the sum of its parts

00:55 We'll substitute appropriate values in the area formula and solve

01:03 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

ABCD is a parallelogram.

CE is its height.

CB = 5
AE = 7
EB = 2

What is the area of the parallelogram?

Step-by-step solution

To find the area,

first, the height of the parallelogram must be found.

To conclude, let's take a look at triangle EBC.

Since we know it is a right triangle (since it is the height of the parallelogram)

the Pythagorean theorem can be used:

$a^2+b^2=c^2$

In this case: $EB^2+EC^2=BC^2$

We place the given information: $2^2+EC^2=5^2$

We isolate the variable: $EC^2=5^2+2^2$

We solve: $EC^2=25-4=21$

$EC=\sqrt{21}$

Now all that remains is to calculate the area.

It is important to remember that for this, the length of each side must be used.
That is, AE+EB=2+7=9

$\sqrt{21}\times9=41.24$

Final Answer

41.24

Key Points to Remember

Essential concepts to master this topic

Height Rule: Use Pythagorean theorem to find perpendicular height CE
Technique: Calculate $2^2 + h^2 = 5^2$ gives h = $\sqrt{21}$
Check: Area = base × height = 9 × $\sqrt{21}$ = 41.24 ✓

Common Mistakes

Avoid these frequent errors

Using the slanted side CB as height instead of perpendicular CE
Don't use CB = 5 as the height directly = wrong area of 45! The slanted side is not perpendicular to the base. Always find the perpendicular distance CE using the Pythagorean theorem with the right triangle EBC.

Practice Quiz

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Calculate the area of the parallelogram based on the data in the figure:

FAQ

Everything you need to know about this question

Why can't I just use CB = 5 as the height?

CB is the slanted side of the parallelogram, not the height! The height must be perpendicular to the base. CE is the perpendicular distance from C to line AB.

How do I know triangle EBC is a right triangle?

CE is described as the height of the parallelogram, which means it's perpendicular to the base AB. This creates a right angle at E, making triangle EBC a right triangle.

What's the base of the parallelogram?

The base is the entire bottom side AB. Since AE = 7 and EB = 2, the total base length is AE + EB = 7 + 2 = 9.

Why do I get a decimal answer instead of a whole number?

Because the height $\sqrt{21}$ is an irrational number (approximately 4.58). When multiplied by 9, you get 41.24. This is perfectly normal in geometry!

How can I check if 41.24 is correct?

Verify that $\sqrt{21} \times 9 = 41.24$ by calculating: $\sqrt{21} \approx 4.583$ , so $4.583 \times 9 \approx 41.24$ ✓

Could I solve this problem differently?

You could use coordinate geometry or trigonometry, but the Pythagorean theorem method is the most straightforward since you're given a right triangle with two known sides.

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Parallelogram Area Calculation: Using Height CE and Segments 5, 7, and 2

Parallelogram Area with Pythagorean Theorem

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just use CB = 5 as the height?

How do I know triangle EBC is a right triangle?

What's the base of the parallelogram?

Why do I get a decimal answer instead of a whole number?

How can I check if 41.24 is correct?

Could I solve this problem differently?

🌟 Unlock Your Math Potential

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