Respuesta :
Check the picture below.
first off we'll need to know the midpoints of ST and UV, well, from ST as you can see in the picture, is just a vertical line, so its midpoint is simply (-2,0), let's find it for UV
[tex]\bf ~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ U(\stackrel{x_1}{3}~,~\stackrel{y_1}{-2})\qquad V(\stackrel{x_2}{13}~,~\stackrel{y_2}{10}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{13+3}{2}~~,~~\cfrac{10-2}{2} \right)\implies \left( \cfrac{16}{2}~~,~~\cfrac{8}{2} \right)\implies (8~~,~~4) \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{-2}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{8}~,~\stackrel{y_2}{4})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d = \sqrt{[8-(-2)]^2+[4-0]^2}\implies d=\sqrt{(8+2)^2+(4-0)^2} \\\\\\ d=\sqrt{10^2+4^2}\implies d=\sqrt{116}\implies d\approx 10.77[/tex]
The required exact length of the midsegment of the trapezoid with the vertices STUV is 11.69.
Given that,
Vertices of trapezoid are; S(−2, 4), T(−2,−4), U(3,−2), V(13, 10).
We have to determine,
The exact length of the midsegment of the trapezoid.
According to the question,
The length of the mid segment of trapezoid is half the sum of the lengths of the two parallel sides, ST and UV.
Vertices of trapezoid are; S(−2, 4), T(−2,−4), U(3,−2), V(13, 10).
Then, The Length of sides ST and UV is determined by the distance formula;
[tex]Length = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex]
The length of side ST is S(−2, 4), T(−2,−4),
[tex]ST = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\\\\ST = \sqrt{((-2)-(-2))^2 + ((-4)-4))^2}}\\\\ST = \sqrt{(0)^2 + (-8)^2}\\\\ST = \sqrt{64}\\\\ST = 8[/tex]
The length of side UV is U(3,−2), V(13, 10).
[tex]ST = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\\\\ST = \sqrt{(13-3)^2 + (10-(-2))^2}}\\\\ST = \sqrt{(10)^2 + (12)^2}\\\\ST = \sqrt{100+144}\\\\ST = \sqrt{244} \\\\ST = 15.62[/tex]
Therefore,
The length of mid segment STUV is,
[tex]Length\ of \ mid \ Segment\ = \dfrac{(ST+UV)}{2}\\\\Length\ of \ mid \ Segment\ = \dfrac{(8+ 15.62)}{2}\\\\Length\ of \ mid \ Segment\ = \dfrac{(23.62)}{2}\\\\Length\ of \ mid \ Segment\ = 11.81[/tex]
Hence, The required exact length of the midsegment of the trapezoid with the vertices STUV is 11.69.
To know more about Trapezoid click the link given below.
https://brainly.com/question/12344210
