Rectangle Side Length: Solve for X When Area = 64 cm² and AB = 4BC

Q: Why do I need to substitute AB = 4x into the area formula?

Because you need one variable to solve! The area formula S = AB × BC has two unknowns, but the relationship AB = 4BC lets you write everything in terms of x.

Q: What does 'AB is 4 times greater than BC' actually mean?

It means AB = 4 × BC. If BC = x, then AB = 4x. So AB is four times as long as BC, not just 4 units longer.

Q: How do I solve 4x² = 64?

First divide both sides by 4 to get $ x^2 = 16 $. Then take the square root of both sides: $ x = 4 $ (we use positive since length is positive).

Q: Should I check both x = 4 and x = -4?

No! Since BC represents a length measurement, it must be positive. Always use x = 4 cm as your final answer.

Q: What if I get the area formula wrong?

Remember that area of a rectangle is always length × width. From the diagram, AB is the length and BC is the width, so Area = AB × BC.

Rectangle Area Problems with Variable Relationships

Given the rectangle ABCD

BC=X and the side AB is 4 times greater than the side BC:

The area of the rectangle is 64 cm².

Calculate the size of the side BC

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:14 Let's find the length of side B C.

00:18 Remember, to calculate the area of a rectangle, multiply side A B by side B C.

00:38 Now, let's plug in the values we know into this equation.

00:45 We're told that side A B is four times longer than side B C.

00:59 So, let's put this into our formula and figure out what X is.

01:27 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the rectangle ABCD

BC=X and the side AB is 4 times greater than the side BC:

The area of the rectangle is 64 cm².

Calculate the size of the side BC

Step-by-step solution

The area of the rectangle equals:

$S=AB\times BC$

$64=AB\times X$

Since it is given that side AB is 4 times larger than side BC

We can state that:

$AB=4BC=4X$

Now let's substitute this information into the formula for calculating the area:

$64=4x\times x$

$64=4x^2$

Let's divide both sides by 4:

$16=x^2$

We'll take the square root as follows:

$4=x$

In other words, BC equals 4

Final Answer

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width always
Variable Relationship: AB = 4BC means substitute 4x for AB
Check: Verify BC = 4 gives area: 4 × 16 = 64 cm² ✓

Common Mistakes

Avoid these frequent errors

Not substituting the relationship correctly into area formula
Don't write 64 = AB × x without replacing AB with 4x = wrong setup! This ignores the given relationship and makes solving impossible. Always substitute AB = 4BC = 4x into the area formula first.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

FAQ

Everything you need to know about this question

Why do I need to substitute AB = 4x into the area formula?

Because you need one variable to solve! The area formula S = AB × BC has two unknowns, but the relationship AB = 4BC lets you write everything in terms of x.

What does 'AB is 4 times greater than BC' actually mean?

It means AB = 4 × BC. If BC = x, then AB = 4x. So AB is four times as long as BC, not just 4 units longer.

How do I solve 4x² = 64?

First divide both sides by 4 to get $x^2 = 16$ . Then take the square root of both sides: $x = 4$ (we use positive since length is positive).

Should I check both x = 4 and x = -4?

No! Since BC represents a length measurement, it must be positive. Always use x = 4 cm as your final answer.

What if I get the area formula wrong?

Remember that area of a rectangle is always length × width. From the diagram, AB is the length and BC is the width, so Area = AB × BC.

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