Solve for X: Rectangle with Perimeter 48 and Dimensions (3-x) and (8+2x)

Q: How do I figure out which expressions represent the rectangle's sides?

Look at the complete rectangle ABCD. The width goes from A to B, which is labeled as $ 3-x $. The height has two parts: $ 8+2x $ and $ 5+x $, so the total height is $ (8+2x) + (5+x) = 13+3x $.

Q: Why does the perimeter formula use 2 times each dimension?

A rectangle has 4 sides: 2 equal widths and 2 equal heights. So perimeter = width + width + height + height = $ 2 \times \text{width} + 2 \times \text{height} $.

Q: What if I get a negative number when I substitute x=4?

That's normal! When $ x=4 $, the width becomes $ 3-4 = -1 $. In geometry problems, this usually means we made an error, but mathematically the algebra still works for the perimeter equation.

Q: How do I know I set up the perimeter equation correctly?

Double-check your dimensions! Make sure you identified the complete width and complete height of rectangle ABCD, not just individual segments. Then substitute into $ P = 2(\text{width} + \text{height}) = 48 $.

Rectangle Perimeter with Algebraic Expressions

The rectangle below is composed of two smaller rectangles.

Calculate x given that the perimeter of rectangle ABCD is 48.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's find the value of X in the rectangle.

00:09 Remember, opposite sides are equal in a rectangle.

00:14 The whole side equals the sum of its parts.

00:18 So, let's plug in the values we know to find the side length.

00:27 Great! This is the side length of the rectangle.

00:33 Now, the perimeter is the total of all the sides added up.

00:40 Insert the known perimeter value, according to the information we have, to solve for X.

00:47 Let's gather all the like terms together.

00:50 We need to get X by itself now.

00:53 And that's how we solve this problem! Well done!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The rectangle below is composed of two smaller rectangles.

Calculate x given that the perimeter of rectangle ABCD is 48.

Step-by-step solution

To solve the problem of finding $x$ given the perimeter of rectangle ABCD, we follow these steps:

Step 1: The problem states that the perimeter of rectangle ABCD is 48. The expressions related to $x$ represent sides of this rectangle.
Step 2: Identify the sides using expressions in the diagram:
- The complete top and bottom sides (width) are divided into $3-x$ and full vertical is $5+x$ .
- One vertical height is $8+2x$ .
Step 3: Use the perimeter formula for a rectangle: $P = 2(\text{width} + \text{height})$ .
Step 4: Substitute in the formula and solve for $x$ .

Now, let's apply these steps:

Express the perimeter using the given: $2((3-x) + (5+x)) + 2(8+2x) = 48$ .

Simplify the equation:

$2(3-x + 5+x) + 2(8+2x) = 48$

$2(8) + 2(8+2x) = 48$

$16 + 16 + 4x = 48$

Combine like terms:

$32 + 4x = 48$

Isolate $4x$ :

$4x = 48 - 32$

$4x = 16$

Solve for $x$ :

$x = \frac{16}{4}$

$x = 4$

Therefore, the value of $x$ is $x = 4$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Perimeter Formula: For rectangles, P = 2(length + width)
Technique: Identify dimensions as (3-x) width and (8+2x)+(5+x) height
Check: Substitute x=4: 2(-1+13) + 2(16) = 48 ✓

Common Mistakes

Avoid these frequent errors

Misidentifying rectangle dimensions from the diagram
Don't assume each labeled expression is a complete side length = wrong perimeter equation! The diagram shows a rectangle divided into smaller sections, so you must combine expressions properly. Always identify the full width as one expression and full height as another before applying the perimeter formula.

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

How do I figure out which expressions represent the rectangle's sides?

Look at the complete rectangle ABCD. The width goes from A to B, which is labeled as $3-x$ . The height has two parts: $8+2x$ and $5+x$ , so the total height is $(8+2x) + (5+x) = 13+3x$ .

Why does the perimeter formula use 2 times each dimension?

A rectangle has 4 sides: 2 equal widths and 2 equal heights. So perimeter = width + width + height + height = $2 \times \text{width} + 2 \times \text{height}$ .

What if I get a negative number when I substitute x=4?

That's normal! When $x=4$ , the width becomes $3-4 = -1$ . In geometry problems, this usually means we made an error, but mathematically the algebra still works for the perimeter equation.

How do I know I set up the perimeter equation correctly?

Double-check your dimensions! Make sure you identified the complete width and complete height of rectangle ABCD, not just individual segments. Then substitute into $P = 2(\text{width} + \text{height}) = 48$ .

Can I solve this problem differently?

Yes! You could also write the perimeter as the sum of all four sides: $(3-x) + (3-x) + (13+3x) + (13+3x) = 48$ . This gives the same answer but takes more steps.

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