Question

Find the length of the midsegment of the trapezoid with the vertices S(0, 0), 7(2, 7), U(6, 10), V(8, 6).

The length of the midsegment is

Answer 1

7.5

Explanation:

A trapezoid is a quadrilateral where the bases are parallel, but the legs are not parallel.

Given vertices of trapezoid STUV:

• S = (0, 0)
• T = (2, 7)
• U = (6, 10)
• V = (8, 6)

The parallel bases of the trapezoid are TU and SV.

Find the lengths of TU and SV using the distance formula.

`\boxed{\begin{minipage}{7.4 cm}\underline{Distance between two points}\\\\$d=√((x_2-x_1)^2+(y_2-y_1)^2)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}`

`\begin{aligned}TU&=√((x_U-x_T)^2+(y_U-y_T)^2)\\&=√((6-2)^2+(10-7)^2)\\&=√((4)^2+(3)^2)\\&=√(16+9)\\&=√(25)\\&=5\end{aligned}`

`\begin{aligned}SV&=√((x_V-x_S)^2+(y_V-y_S)^2)\\&=√((8-0)^2+(6-0)^2)\\&=√((8)^2+(6)^2)\\&=√(64+36)\\&=√(100)\\&=10\end{aligned}`

The midsegment of a trapezoid is the segment parallel to the bases that connects the midpoints of the legs.

`\boxed{\begin{minipage}{6.5cm}\underline{Midsegment of a Trapezoid}\\\\$M=(b_1+b_2)/(2)$\\\\where:\\ \phantom{ww}$\bullet$ $M$ is the length of the midsegment. \\\phantom{ww}$\bullet$ $b_1$ and $b_2$ are the parallel bases. \\\end{minipage}}`

Substitute the found lengths of the bases into the midsegment formula:

`\implies M=(5+10)/(2)`

`\implies M=(15)/(2)`

`\implies M=7.5`

The length of the midsegment is 7.5.