Question

s and v are the midpoints of the legs, $overline{rt}$ and $overline{qu}$, of trapezoid $qrtu$. if $qr = 2x - 68$, $sv = x$, and $tu = 2x - 32$, what is the value of $x$?

Explanation:

Step1: Recall mid - segment formula

The length of the mid - segment of a trapezoid (the line segment connecting the midpoints of the non - parallel sides) is given by the formula $SV=\frac{QR + TU}{2}$.

Step2: Substitute given expressions

Substitute $QR = 2x-68$, $SV=x$, and $TU = 2x - 32$ into the formula: $x=\frac{(2x - 68)+(2x - 32)}{2}$.

Step3: Simplify the right - hand side

First, simplify the numerator: $(2x - 68)+(2x - 32)=2x+2x-68 - 32=4x-100$. So the equation becomes $x=\frac{4x - 100}{2}$.

Step4: Cross - multiply

Multiply both sides of the equation by 2: $2x=4x - 100$.

Step5: Solve for x

Subtract $4x$ from both sides: $2x-4x=-100$. Combine like terms: $-2x=-100$. Divide both sides by $-2$: $x = 50$.

Answer:

$50$