Question

use the figure below for questions 9 - 11. 9. if de = 4x - 1, ef = 9, and df = 9x - 22, find the value of x. 10. if df = 78, de = 5x - 9, and ef = 2x + 10, find ef. 11. if de = 4x + 10, ef = 2x - 1, and df = 9x - 15, find df.

Explanation:

Step1: Use segment - addition postulate

Since $DF = DE+EF$ for collinear points $D$, $E$, and $F$.

Step2: Solve for $x$ in question 9

Given $DE = 4x - 1$, $EF=9$, and $DF = 9x - 22$. Substitute into $DF = DE + EF$:
\[9x-22=(4x - 1)+9\]
\[9x-22=4x+8\]
\[9x-4x=8 + 22\]
\[5x=30\]
\[x = 6\]

Step3: Solve for $EF$ in question 10

Given $DF = 78$, $DE = 5x - 9$, and $EF = 2x+10$. Substitute into $DF=DE + EF$:
\[78=(5x - 9)+(2x + 10)\]
\[78=5x-9+2x + 10\]
\[78=7x+1\]
\[7x=78 - 1\]
\[7x=77\]
\[x = 11\]
Then $EF=2x + 10=2\times11+10=22 + 10=32$

Step4: Solve for $DF$ in question 11

Given $DE = 4x + 10$, $EF = 2x-1$, and $DF = 9x-15$. Substitute into $DF=DE + EF$:
\[9x-15=(4x + 10)+(2x-1)\]
\[9x-15=4x+10+2x - 1\]
\[9x-15=6x + 9\]
\[9x-6x=9 + 15\]
\[3x=24\]
\[x = 8\]
Then $DF=9x-15=9\times8-15=72-15 = 57$

Answer:

1. $x = 6$
2. $EF = 32$
3. $DF = 57$