Rectangle Perimeter Problem: Solving for X When One Side is 2x

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

A rectangle has 4 sides: two lengths and two widths. The formula P = 2(l + w) accounts for both pairs of opposite sides being equal.

Q: What if the rectangle shows different variables on each side?

Look carefully at the diagram! In rectangles, opposite sides are always equal. If one side is labeled 2x, the opposite side is also 2x, even if not labeled.

Perimeter Equations with Algebraic Dimensions

The perimeter of the rectangle below is 12.

Calculate x.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Opposite sides are equal in a rectangle

00:21 The perimeter of the rectangle equals the sum of its sides

00:32 We'll substitute appropriate values and solve for X

00:48 Collect like terms

01:01 Isolate X

01:10 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

The perimeter of the rectangle below is 12.

Calculate x.

Step-by-step solution

To solve this problem, we will determine $x$ using the perimeter formula for a rectangle. The steps are as follows:

Step 1: Set up the perimeter equation for this problem using the formula $P = 2(l + w)$ .
Step 2: Identify the given dimensions: $l = 2x$ and $w = x$ .
Step 3: Substitute these dimensions into the perimeter equation: $2(2x + x) = 12$ .
Step 4: Simplify the equation to solve for $x$ .

Now, let's follow these steps:

Step 1: The perimeter $P = 12$ units. Thus, we use the equation:

2(l + w) = 12

Step 2: Substitute the known values of length and width:

2(2x + x) = 12

Step 3: Simplify the equation inside the parentheses:

2(3x) = 12

Step 4: Divide both sides by 2 to solve for $x$ :

3x = 6

Finally, divide by 3:

x = 2

Therefore, the solution to the problem is $x = 2$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Rectangle perimeter equals 2 times length plus width
Technique: Substitute known values: 2(2x + x) = 12
Check: Verify by calculating: 2(4 + 2) = 12 ✓

Common Mistakes

Avoid these frequent errors

Adding dimensions instead of using perimeter formula
Don't just add 2x + x = 12 to solve for x = 4! This ignores that rectangles have four sides, not two. Always use P = 2(l + w) which accounts for all four sides of the rectangle.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

Why do I multiply by 2 in the perimeter formula?

A rectangle has 4 sides: two lengths and two widths. The formula P = 2(l + w) accounts for both pairs of opposite sides being equal.

What if the rectangle shows different variables on each side?

Look carefully at the diagram! In rectangles, opposite sides are always equal. If one side is labeled 2x, the opposite side is also 2x, even if not labeled.

How do I know which dimension is length vs width?

It doesn't matter for perimeter problems! Whether you call it length × width or width × length, the perimeter formula P = 2(l + w) gives the same result.

Can I solve this problem without the perimeter formula?

You could add all four sides: 2x + x + 2x + x = 12, but using P = 2(l + w) is faster and reduces calculation errors!

What if I get a decimal or fraction for x?

That's perfectly normal! Not all rectangle problems have whole number solutions. Always double-check by substituting your answer back into the original equation.

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