Question

Find the length of the median of each trapezoid.
top left corner F
Left middle U
Left bottom corner G
Right top corner E
right Middle V
Right bottom corner D
Top 2x+4
bottom 2x+18
center 5x-16


Question 7 options:

18


34


28


29

Answer 1

To find the length of the median of the trapezoid defined by the vertices \( F, U, G, E, V, D \) with the provided top and bottom bases, we will use the formula for the length of the median (or midsegment) of a trapezoid:

\[
m = \frac{b_1 + b_2}{2}
\]

where \( b_1 \) is the length of the top base and \( b_2 \) is the length of the bottom base.

From the question, we have:
- The top base (length \( b_1 \)) is given as \( 2x + 4 \).
- The bottom base (length \( b_2 \)) is given as \( 2x + 18 \).

Now let’s calculate the length of the median \( m \):

\[
m = \frac{(2x + 4) + (2x + 18)}{2}
\]

Now simplify the expression:

\[
m = \frac{2x + 4 + 2x + 18}{2}
\]
\[
m = \frac{4x + 22}{2}
\]
\[
m = 2x + 11
\]

Next, we also need to determine the value of \( x \) using the center length of the trapezoid which is given as \( 5x - 16 \). In a trapezoid, the median should also equal the average of the two bases. Thus, we set:

\[
2x + 11 = 5x - 16
\]

Now we can solve for \( x \):

\[
11 + 16 = 5x - 2x
\]
\[
27 = 3x
\]
\[
x = 9
\]

Now substituting \( x = 9 \) back into the equation for median \( m \):

\[
m = 2(9) + 11 = 18 + 11 = 29
\]

Thus, the length of the median of the trapezoid is:

\[
\boxed{29}
\]