Find Side BC in Parallelogram: Perimeter = 30 cm with Side CD = 2x

Q: Why can I use only two different side lengths for a parallelogram?

Opposite sides are always equal in a parallelogram! So if $ CD = 2x $, then $ AB = 2x $ too. This means we only need to find two different side lengths, not four.

Q: How do I know which sides to call 'a' and 'b' in the formula?

It doesn't matter! You can call $ BC $ either 'a' or 'b'. The key is using one pair of opposite sides. Since $ CD = 2x $ is given, pair it with $ BC $.

Q: Can I solve this without using the perimeter formula?

You could write $ AB + BC + CD + AD = 30 $, but then you'd substitute $ AB = CD = 2x $ and $ AD = BC $ anyway. The perimeter formula saves time by using the parallelogram property directly!

Q: How do I check my answer when x is still a variable?

Substitute $ BC = 15-2x $ back into the perimeter formula: $ 2(15-2x) + 2(2x) = 30-4x+4x = 30 $ ✓. The x terms cancel out, confirming our answer works for any valid x value!

Parallelogram Perimeter with Variable Expressions

How long is side BC given that the perimeter of the parallelogram is 30 cm?

$CD=2x$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find BC

00:04 The perimeter of the parallelogram equals the sum of the sides

00:07 The perimeter of the parallelogram equals the sum of the sides

00:11 We'll substitute appropriate values according to the given data, and solve for BC

00:20 We'll isolate BC

00:35 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

How long is side BC given that the perimeter of the parallelogram is 30 cm?

$CD=2x$

Step-by-step solution

To solve this problem, we begin by using the formula for the perimeter of a parallelogram:

The perimeter $P$ is given by: $P = 2(a + b)$

Given: $P = 30 \text{ cm}$ , and $CD = 2x$ . We know $AB = 2x$ since opposite sides of a parallelogram are equal. So, we write:

$P = 2(BC + CD) = 30$
$2(BC + 2x) = 30$
$BC + 2x = 15$ (after dividing both sides by 2)
$BC = 15 - 2x$

Thus, the length of side $BC$ is given by:

$BC = 15 - 2x$

Therefore, the correct option is:

Choice 1: $15 - 2x$

This matches the problem's given correct answer.

Final Answer

$15-2x$

Key Points to Remember

Essential concepts to master this topic

Parallelogram Property: Opposite sides are equal in length
Technique: Use $P = 2(BC + CD)$ with perimeter 30 cm
Check: Verify $2(15-2x) + 2(2x) = 30$ simplifies correctly ✓

Common Mistakes

Avoid these frequent errors

Using all four sides instead of recognizing opposite sides are equal
Don't write $AB + BC + CD + DA = 30$ = four unknown expressions! This makes the problem unsolvable because you can't find one variable with four different expressions. Always use $P = 2(a + b)$ since opposite sides are equal.

Practice Quiz

Test your knowledge with interactive questions

Given the parallelogram:

Calculate the perimeter of the parallelogram.

FAQ

Everything you need to know about this question

Why can I use only two different side lengths for a parallelogram?

Opposite sides are always equal in a parallelogram! So if $CD = 2x$ , then $AB = 2x$ too. This means we only need to find two different side lengths, not four.

How do I know which sides to call 'a' and 'b' in the formula?

It doesn't matter! You can call $BC$ either 'a' or 'b'. The key is using one pair of opposite sides. Since $CD = 2x$ is given, pair it with $BC$ .

What if I get a negative answer for BC?

That's possible depending on the value of x! Side lengths must be positive, so this tells you there are restrictions on what x can be. For example, if $BC = 15-2x$ , then $x < 7.5$ .

Can I solve this without using the perimeter formula?

You could write $AB + BC + CD + AD = 30$ , but then you'd substitute $AB = CD = 2x$ and $AD = BC$ anyway. The perimeter formula saves time by using the parallelogram property directly!

How do I check my answer when x is still a variable?

Substitute $BC = 15-2x$ back into the perimeter formula: $2(15-2x) + 2(2x) = 30-4x+4x = 30$ ✓. The x terms cancel out, confirming our answer works for any valid x value!

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