Question

x and u are the midpoints of the legs, \\(\overline{wy}\\) and \\(\overline{vz}\\), of trapezoid vwyz. if \\(yz = p - 73\\), \\(ux = -p + 94\\), and \\(vw = -p + 97\\), what is the value of \\(p\\)?

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. Its length is the average of the lengths of the two bases. So, \( UX=\frac{YZ + VW}{2} \).

Step2: Substitute the given expressions into the formula

We know \( YZ = p - 73 \), \( UX=-p + 94 \), and \( VW=-p + 97 \). Substituting these into the formula \( UX=\frac{YZ + VW}{2} \), we get:

$$ -p + 94=\frac{(p - 73)+(-p + 97)}{2} $$

Step3: Simplify the right - hand side of the equation

First, simplify the numerator of the fraction on the right - hand side:
\((p - 73)+(-p + 97)=p - 73 - p+97=(p - p)+(97 - 73)=24\)
So the equation becomes:

$$ -p + 94=\frac{24}{2} $$

Since \(\frac{24}{2}=12\), the equation is:

$$ -p + 94 = 12 $$

Step4: Solve for \( p \)

Subtract 94 from both sides of the equation:

$$ -p=12 - 94 $$

$$ -p=-82 $$

Multiply both sides by - 1:

$$ p = 82 $$

Answer:

\( p = 82 \)