Answer
$AB = 6$
$CD = 10$
$EF = 8$
Work Step by Step
According to the trapezoid midsegment theorem, in a quadrilateral that is a trapezoid, the midsegment is parallel to the bases and is half the sum of the base lengths.
Let's set up the equation to find the value of $x$:
$EF = \frac{x}{y}(AB + CD)$
Let's plug in what we are given:
$4x = \frac{1}{2}[(5x - 4) + (6x - 2)]$
Evaluate parentheses first:
$4x = \frac{1}{2}(11x - 6)$
Divide both sides by $\frac{1}{2}$ to get rid of the fraction.
$2(4x) = 11x - 6$
Multiply on the left side of the equation to simplify:
$8x = 11x - 6$
Subtract $11x$ from both sides of the equation to move variable terms to the left side of the equation:
$-3x = -6$
Divide both sides by $-3$ to solve for $x$:
$x = 2$
Now we plug $2$ in for $x$.
Let's look at the expression for the shorter base:
$AB = 5x - 4$
Let's substitute $2$ for $x$:
$AB = 5(2) - 4$
Multiply first, according to order of operations:
$AB = 10 - 4$
Subtract to solve:
$AB = 6$
Let's look at the expression for the longer base:
$CD = 6x - 2$
Substitute $2$ for $x$ into the equation:
$CD = 6(2) - 2$
Multiply first, according to order of operations:
$CD = 12 - 2$
Subtract to solve:
$CD = 10$
Finally, let's look at the expression for the midsegment:
$EF = 4x$
Substitute $2$ for $x$:
$EF = 4(2)$
Multiply to solve:
$EF = 8$
