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

Test your knowledge with interactive questions

Look at the triangle in the diagram. How long is side AB?

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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