Question

x and u are the midpoints of the legs, \\(\overline{sw}\\) and \\(\overline{tv}\\), of trapezoid stvw. if \\(vw = -7y + 90\\), \\(ux = -2y + 73\\), and \\(st = 10y + 14\\), what is the value of \\(y\\)?

Explanation:

Step1: Recall the midsegment theorem for trapezoids.

The midsegment (or median) of a trapezoid is the segment that connects the midpoints of the legs, and its length is the average of the lengths of the two bases. So, for trapezoid \(STVW\) with bases \(ST\) and \(VW\), and midsegment \(UX\), we have the formula:
\(UX=\frac{ST + VW}{2}\)

Step2: Substitute the given expressions into the formula.

We know \(VW=-7y + 90\), \(UX=-2y + 73\), and \(ST = 10y + 14\). Substituting these into the midsegment formula:
\(-2y + 73=\frac{(10y + 14)+(-7y + 90)}{2}\)

Step3: Simplify the right-hand side.

First, combine like terms in the numerator:
\((10y + 14)+(-7y + 90)=10y-7y + 14 + 90 = 3y + 104\)
So the equation becomes:
\(-2y + 73=\frac{3y + 104}{2}\)

Step4: Eliminate the fraction by multiplying both sides by 2.

\(2(-2y + 73)=3y + 104\)
Simplify the left-hand side:
\(-4y + 146 = 3y + 104\)

Step5: Solve for \(y\) by moving all \(y\)-terms to one side and constants to the other.

Add \(4y\) to both sides:
\(146 = 7y + 104\)
Subtract 104 from both sides:
\(146 - 104 = 7y\)
\(42 = 7y\)

Step6: Divide both sides by 7.

\(y=\frac{42}{7}=6\)

Answer:

\(y = 6\)