Solve for X: Trapezoid Perimeter of 78cm with Side Lengths 2X, X+3, and 4X-8

Q: Why do we set up the equation as 78 = X + (4X-8) + 2X + (X+3)?

The perimeter is the total distance around the trapezoid, so we add all four side lengths. Each side has a different expression: AB = X, BC = 4X-8, CD = 2X, and DA = X+3.

Q: How do I know which terms to combine?

Look for terms with the same variable! All X terms go together: X + 4X + 2X + X = 8X. Then combine the constants: -8 + 3 = -5. So you get 8X - 5.

Q: What if I get a fraction as my answer?

Fractions are perfectly valid answers! $ X = \frac{83}{8} $ is the exact answer. You can also write it as a mixed number: $ 10\frac{3}{8} $, but the fraction form is usually preferred.

Q: How can I check if my answer is correct?

Substitute $ X = \frac{83}{8} $ into each side length, then add them up. If the sum equals 78 cm, you're correct! For example: $ \frac{83}{8} + (4 \cdot \frac{83}{8} - 8) + 2 \cdot \frac{83}{8} + (\frac{83}{8} + 3) = 78 $

Q: Why didn't we just guess and check instead?

Setting up an equation is more reliable than guessing! With algebra, you get the exact answer in just a few steps. Guessing could take forever and you might never find $ \frac{83}{8} $!

Perimeter Equations with Variable Expressions

The perimeter of the trapezoid ABCD is equal to 78 cm. Calculate X.

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Understand the problem

The perimeter of the trapezoid ABCD is equal to 78 cm. Calculate X.

Step-by-step solution

To solve this problem, we will set up an equation using the given expressions and the perimeter formula.

Let's denote the side lengths of trapezoid $ABCD$ as follows:

$AB = X$
$BC = 4X - 8$
$CD = 2X$
$DA = X + 3$

The perimeter of trapezoid $ABCD$ is given by the sum of its sides:

$\text{Perimeter} = AB + BC + CD + DA$

Substitute the expressions for the side lengths into the perimeter formula:

$78 = X + (4X - 8) + 2X + (X + 3)$

Combine like terms:

$78 = X + 4X - 8 + 2X + X + 3$

$78 = 8X - 5$

Add 5 to both sides to isolate terms with $X$ :

$78 + 5 = 8X$

$83 = 8X$

Divide both sides by 8 to solve for $X$ :

$X = \frac{83}{8}$

Thus, the value of $X$ is $\frac{83}{8}$ , which corresponds to choice 1: $x=\frac{83}{8}$ .

Final Answer

$x=\frac{83}{8}$

Key Points to Remember

Essential concepts to master this topic

Perimeter Formula: Add all four sides of the trapezoid together
Technique: Combine like terms: X + 4X + 2X + X = 8X
Check: Substitute $X = \frac{83}{8}$ back into perimeter equation: 78 = 78 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to combine like terms correctly
Don't just add the coefficients without tracking the variable terms = wrong equation setup! Students often write 2X + X + 3 + 4X - 8 instead of grouping X terms together first. Always collect all X terms: (2X + X + X + 4X) + (3 - 8) = 8X - 5.

Practice Quiz

Test your knowledge with interactive questions

Calculate the perimeter of the trapezoid according to the following data:

FAQ

Everything you need to know about this question

Why do we set up the equation as 78 = X + (4X-8) + 2X + (X+3)?

The perimeter is the total distance around the trapezoid, so we add all four side lengths. Each side has a different expression: AB = X, BC = 4X-8, CD = 2X, and DA = X+3.

How do I know which terms to combine?

Look for terms with the same variable! All X terms go together: X + 4X + 2X + X = 8X. Then combine the constants: -8 + 3 = -5. So you get 8X - 5.

What if I get a fraction as my answer?

Fractions are perfectly valid answers! $X = \frac{83}{8}$ is the exact answer. You can also write it as a mixed number: $10\frac{3}{8}$ , but the fraction form is usually preferred.

How can I check if my answer is correct?

Substitute $X = \frac{83}{8}$ into each side length, then add them up. If the sum equals 78 cm, you're correct! For example: $\frac{83}{8} + (4 \cdot \frac{83}{8} - 8) + 2 \cdot \frac{83}{8} + (\frac{83}{8} + 3) = 78$

Why didn't we just guess and check instead?

Setting up an equation is more reliable than guessing! With algebra, you get the exact answer in just a few steps. Guessing could take forever and you might never find $\frac{83}{8}$ !

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