Question

point e is on line segment $overline{df}$. given $de = 2x$, $ef = 2x - 6$, and $df = 3x + 5$, determine the numerical length of $overline{ef}$.

Explanation:

Step1: Use segment - addition postulate

Since point E is on line segment $\overline{DF}$, we have $DE + EF=DF$. Substitute the given expressions: $2x+(2x - 6)=3x + 5$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $2x+2x-6 = 4x-6$. So the equation becomes $4x-6 = 3x + 5$.

Step3: Solve for x

Subtract $3x$ from both sides: $4x-3x-6=3x-3x + 5$, which simplifies to $x-6 = 5$. Then add 6 to both sides: $x=5 + 6$, so $x = 11$.

Step4: Find the length of EF

Substitute $x = 11$ into the expression for $EF$. Since $EF=2x-6$, then $EF=2\times11-6$. First, calculate $2\times11 = 22$, then $22-6=16$.

Answer:

16