Question

u and r are the midpoints of the legs, \\(\overline{pt}\\) and \\(\overline{qs}\\), of trapezoid pqst. if \\(st = -8w + 26\\), \\(ru = -w + 24\\), and \\(pq = 3w + 25\\), what is the value of w?

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, \( RU=\frac{PQ + ST}{2} \).

Step2: Substitute the given expressions into the formula.

We know \( ST=-8w + 26 \), \( RU=-w + 24 \), and \( PQ = 3w+25 \). Substituting these into the formula \( RU=\frac{PQ + ST}{2} \), we get:
\[
-w + 24=\frac{(3w + 25)+(-8w + 26)}{2}
\]

Step3: Simplify the right - hand side numerator.

First, combine like terms in the numerator: \( (3w+25)+(-8w + 26)=3w-8w+25 + 26=-5w + 51 \). So the equation becomes:
\[
-w + 24=\frac{-5w + 51}{2}
\]

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

Multiply both sides of the equation by 2: \( 2(-w + 24)=-5w + 51 \).
Using the distributive property on the left - hand side: \( -2w+48=-5w + 51 \).

Step5: Solve for \( w \).

Add \( 5w \) to both sides of the equation: \( -2w + 5w+48=-5w+5w + 51 \), which simplifies to \( 3w+48 = 51 \).
Then subtract 48 from both sides: \( 3w+48 - 48=51 - 48 \), so \( 3w=3 \).
Divide both sides by 3: \( w = 1 \).

Answer:

\( w = 1 \)