What is the length of each segment in a trapezoid tuvw?

Step-by-step explanation:

we know that

In this problem

XY is the mid-segment of trapezoid TUVW (because K is the midpoint segment TU and Y is the midpoint segment VW)

Remember that the length of the mid-segment is the sum of the two parallel bases divided by 2

so

substitute the given values

solve for x

Multiply by 2 both sides

Find the length segment UV

substitute the value of x

Find the length segment XY

substitute the value of x

Find the length segment TW

substitute the value of x

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4