Parallelogram Perimeter: Finding the Expression When One Side is 4X

Q: How do I know which sides are the long ones and which are short?

The problem tells you the longest sides are X cm. Since a parallelogram has two pairs of equal opposite sides, you have two long sides of X cm each and two short sides.

Q: Why is the short side X/4 and not X-4?

The problem says the long side is four times longer than the short side. This means: $ \text{long side} = 4 \times \text{short side} $. So if long side = X, then short side = X/4.

Q: How do I add X + X/4 + X + X/4?

Group like terms: $ 2X + 2 \cdot \frac{X}{4} = 2X + \frac{2X}{4} = 2X + \frac{X}{2} $. Convert to same denominator: $ \frac{4X}{2} + \frac{X}{2} = \frac{5X}{2} = 2.5X $

Q: What if I got a different answer like 3X or 5X?

Check your work! If you got 3X, you might have used X + X + X = 3X (forgetting the short sides). If you got 5X, you might have calculated 4X + X. Remember: two long sides + two short sides = 2X + 2(X/4) = 2.5X.

Parallelogram Properties with Variable Side Relationships

The longest sides of a parallelogram are X cm long and are four times longer than the shorter sides.

Express the perimeter of the parallelogram in terms of X.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's find the perimeter of the parallelogram using X.

00:15 Remember, in a parallelogram, opposite sides are always equal.

00:25 Here, we're looking at the longer sides of the parallelogram.

00:34 According to the problem, we have a ratio for the sides.

00:53 We'll call side A B as X.

01:07 Next, we'll use this value of A B to find side A C.

01:15 Remember, the perimeter is the total of all sides combined.

01:26 Let's plug in the values and calculate the perimeter.

01:39 And that's how we solve the problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The longest sides of a parallelogram are X cm long and are four times longer than the shorter sides.

Express the perimeter of the parallelogram in terms of X.

Step-by-step solution

In a parallelogram, each pair of opposite sides are equal and parallel: AB = CD and AC = BD.

Given that the length of one side is 4 times greater than the other side equal to X, we know that:

$AB=CD=4AC=4BD$

Now we replace the data in this equation with out own (assuming that AB = CD = X):

$x=x=4AC=4BD$

We divide by 4:

$\frac{x}{4}=\frac{x}{4}=AC=BD$

Now we calculate the perimeter of the parallelogram and express both AC and BD using X:

$P=x+\frac{x}{4}+x+\frac{x}{4}$

$P=2x+\frac{x}{4}+\frac{x}{4}=2\frac{1}{2}x$

Final Answer

2.5X cm

Key Points to Remember

Essential concepts to master this topic

Parallelogram Rule: Opposite sides are equal in length
Technique: If long side = X, then short side = X/4
Check: Perimeter = 2(long) + 2(short) = 2X + 2(X/4) = 2.5X ✓

Common Mistakes

Avoid these frequent errors

Adding all sides as X without considering the ratio
Don't assume all sides equal X and calculate 4X as perimeter = wrong answer! The problem states one side is 4 times longer than the other. Always identify which sides are long (X) and which are short (X/4) first.

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

How do I know which sides are the long ones and which are short?

The problem tells you the longest sides are X cm. Since a parallelogram has two pairs of equal opposite sides, you have two long sides of X cm each and two short sides.

Why is the short side X/4 and not X-4?

The problem says the long side is four times longer than the short side. This means: $\text{long side} = 4 \times \text{short side}$ . So if long side = X, then short side = X/4.

Can I use different letters for the sides?

Yes! You could call the short side 's' and long side '4s'. Then perimeter = 2s + 2(4s) = 10s. But since the problem gives the long side as X, it's easier to use long = X and short = X/4.

How do I add X + X/4 + X + X/4?

Group like terms: $2X + 2 \cdot \frac{X}{4} = 2X + \frac{2X}{4} = 2X + \frac{X}{2}$ . Convert to same denominator: $\frac{4X}{2} + \frac{X}{2} = \frac{5X}{2} = 2.5X$

What if I got a different answer like 3X or 5X?

Check your work! If you got 3X, you might have used X + X + X = 3X (forgetting the short sides). If you got 5X, you might have calculated 4X + X. Remember: two long sides + two short sides = 2X + 2(X/4) = 2.5X.

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