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

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