Question

sv is the midsegment of the trapezoid tuwx. if tu = -10p + 96, sv = p, and wx = -9p + 93, what is the value of p?

Explanation:

Step1: Recall midsegment formula for trapezoid

The midsegment \( SV \) of trapezoid \( TUWX \) is equal to the average of the two bases \( TU \) and \( WX \). So the formula is \( SV=\frac{TU + WX}{2} \).

Step2: Substitute the given expressions

We know \( TU=- 10p + 96 \), \( SV = p \), and \( WX=-9p + 93 \). Substitute these into the formula:
\( p=\frac{(-10p + 96)+(-9p + 93)}{2} \)

Step3: Simplify the right - hand side

First, simplify the numerator: \( (-10p + 96)+(-9p + 93)=-10p-9p + 96 + 93=-19p+189 \)
So the equation becomes \( p=\frac{-19p + 189}{2} \)
Multiply both sides by 2 to get rid of the denominator: \( 2p=-19p + 189 \)

Step4: Solve for \( p \)

Add \( 19p \) to both sides of the equation: \( 2p+19p=-19p + 189+19p \)
\( 21p = 189 \)
Divide both sides by 21: \( p=\frac{189}{21}=9 \)

Answer:

\( p = 9 \)