Solve for X: Parallelogram with Perimeter 20 and Side Ratio 1:3

Q: Why do we multiply by 2 in the perimeter formula?

A parallelogram has four sides, but only two different lengths. Since opposite sides are equal, we have two sides of length x and two sides of length 3x, so P = 2x + 2(3x) = 2(x + 3x).

Q: What if I set up the equation wrong?

Always check your setup first! Make sure you're using P = 2(sum of different side lengths), not just adding the given expressions. Then solve step by step.

Perimeter Equations with Variable Expressions

Calculate X in the parallelogram shown below.

P = Perimeter

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Opposite sides are equal in a parallelogram

00:12 The perimeter of the parallelogram equals the sum of its sides

00:30 Substitute appropriate values and solve for X

00:37 Isolate X

00:46 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

Calculate X in the parallelogram shown below.

P = Perimeter

Step-by-step solution

To calculate $x$ in the given parallelogram, we begin by noting the given perimeter formula for the parallelogram, $P = 2(a + b)$ .

The two sides, as given, are $a = x$ and $b = 3x$ . Substituting these into the formula gives:

$P = 2(x + 3x)$

We know the perimeter $P = 20$ , therefore:

$2(x + 3x) = 20$

Simplifying inside the parentheses, $x + 3x = 4x$ , we rewrite the equation as:

$2 \times 4x = 20$

Which gives:

$8x = 20$

To solve for $x$ , divide both sides by 8:

$x = \frac{20}{8}$

Simplifying gives:

$x = \frac{5}{2}$

Or, $x = 2.5$ .

Thus, the value of $x$ is $\boxed{2\frac{1}{2}}$ .

Final Answer

$2\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Perimeter Formula: For parallelograms, P = 2(length + width)
Technique: Substitute x and 3x into P = 2(x + 3x) = 20
Check: Verify 2.5 works: 2(2.5 + 7.5) = 2(10) = 20 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to multiply by 2 in perimeter formula
Don't use P = x + 3x = 20, getting x = 5! This ignores that a parallelogram has two pairs of opposite sides. Always use P = 2(x + 3x) to account for all four sides.

Practice Quiz

Test your knowledge with interactive questions

Find the perimeter of the parallelogram using the data below.

FAQ

Everything you need to know about this question

Why do we multiply by 2 in the perimeter formula?

A parallelogram has four sides, but only two different lengths. Since opposite sides are equal, we have two sides of length x and two sides of length 3x, so P = 2x + 2(3x) = 2(x + 3x).

How do I know which sides are x and which are 3x?

Look at the diagram! The labels show one pair of opposite sides is x and the other pair is 3x. In parallelograms, opposite sides are always equal.

Can x be a fraction or decimal?

Absolutely! Many geometry problems have fractional answers. $x = 2\frac{1}{2}$ or 2.5 is perfectly valid for a side length.

What if I set up the equation wrong?

Always check your setup first! Make sure you're using P = 2(sum of different side lengths), not just adding the given expressions. Then solve step by step.

How can I verify my answer is correct?

Substitute back: if x = 2.5, then the sides are 2.5 and 7.5. Check: P = 2(2.5 + 7.5) = 2(10) = 20 ✓. Always verify with the original perimeter!

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