Question

find df if d is between e and f, ed = 6x - 4, df = 3x + 5, and ef = 46.

Explanation:

Step1: Use the segment - addition postulate

Since \(D\) is between \(E\) and \(F\), we have \(ED + DF=EF\).
Given \(ED = 6x - 4\), \(DF = 3x + 5\), and \(EF = 46\). Substitute these into the equation: \((6x - 4)+(3x + 5)=46\).

Step2: Simplify the left - hand side of the equation

Combine like terms: \(6x-4 + 3x+5=(6x + 3x)+(-4 + 5)=9x+1\). So the equation becomes \(9x + 1=46\).

Step3: Solve for \(x\)

Subtract \(1\) from both sides: \(9x+1 - 1=46 - 1\), which gives \(9x=45\).
Divide both sides by \(9\): \(x=\frac{45}{9}=5\).

Step4: Find the value of \(DF\)

Substitute \(x = 5\) into the expression for \(DF\). Since \(DF = 3x+5\), then \(DF=3\times5 + 5\).
Calculate \(3\times5+5=15 + 5=20\).

Answer:

\(20\)