Question

fill in the blank 18 points
trapezoid stvu is shown below. find the length of tv if wx is the midsegment of the trapezoid and tv = 3x + 20.
tv = type your answer...

Explanation:

Step1: Recall trapezoid midsegment formula

The midsegment of a trapezoid is the average of the two bases. So, \( WX=\frac{ST + UV}{2} \). Given \( ST = 10x + 5 \), \( UV=8x - 1 \), and \( WX = 7x + 8 \). Substitute into the formula:
\[ 7x + 8=\frac{(10x + 5)+(8x - 1)}{2} \]

Step2: Simplify the right - hand side

First, simplify the numerator: \( (10x + 5)+(8x - 1)=10x+8x + 5 - 1=18x + 4 \). Then the equation becomes:
\[ 7x + 8=\frac{18x + 4}{2} \]
Simplify the right - hand side: \( \frac{18x + 4}{2}=9x + 2 \). So the equation is \( 7x+8 = 9x + 2 \).

Step3: Solve for x

Subtract \( 7x \) from both sides: \( 8=2x + 2 \). Subtract 2 from both sides: \( 6 = 2x \). Divide both sides by 2: \( x = 3 \).

Step4: Find the length of TV

Given \( TV=3x + 20 \), substitute \( x = 3 \): \( TV=3\times3+20=9 + 20=29 \).

Answer:

29