To solve the problem, we need to use the information given:
1. [tex]\( E \)[/tex] is the midpoint of [tex]\( \overline{DF} \)[/tex].
2. Thus, [tex]\( DE = EF \)[/tex].
3. We are given that [tex]\( DE = 4x \)[/tex] and [tex]\( EF = x + 6 \)[/tex].
Since [tex]\( DE = EF \)[/tex], we can set up the equation:
[tex]\[ 4x = x + 6 \][/tex]
Next, we solve for [tex]\( x \)[/tex]. To do this, we'll get all terms involving [tex]\( x \)[/tex] on one side of the equation and constant terms on the other side:
[tex]\[ 4x - x = 6 \][/tex]
[tex]\[ 3x = 6 \][/tex]
Now, divide both sides by 3 to isolate [tex]\( x \)[/tex]:
[tex]\[ x = \frac{6}{3} \][/tex]
[tex]\[ x = 2 \][/tex]
Having found the value of [tex]\( x \)[/tex], we substitute it back into the expression for [tex]\( EF \)[/tex] to determine its length:
[tex]\[ EF = x + 6 \][/tex]
[tex]\[ EF = 2 + 6 \][/tex]
[tex]\[ EF = 8 \][/tex]
So, the length of [tex]\( EF \)[/tex] is [tex]\( 8 \)[/tex].
Thus, [tex]\( EF \)[/tex] is [tex]\( 8 \)[/tex].
