Question

m is the mid - point of each segment. set up an equation to find x and then find the specified lengths.
10.
11.
for the first segment: (7x - 6) (length of em), (5x) (length of mf)
for the second segment: (6x + 11) (length of jm), (9x - 13) (length of mk)

Explanation:

Step1: Set up equation using mid - point property

Since $M$ is the mid - point of $\overline{EF}$, then $EM = MF$. So, $7x−6 = 5x$.

Step2: Solve the equation for $x$

Subtract $5x$ from both sides: $7x−5x−6=5x - 5x$, which simplifies to $2x−6 = 0$. Then add 6 to both sides: $2x−6 + 6=0 + 6$, so $2x=6$. Divide both sides by 2: $\frac{2x}{2}=\frac{6}{2}$, and $x = 3$.

Step3: Find the length of $EM$ and $MF$

Substitute $x = 3$ into the expression for $EM$ (or $MF$). $EM=7x−6=7\times3−6=21 - 6=15$, $MF = 5x=5\times3 = 15$.

For the second segment:

Step1: Set up equation using mid - point property

Since $M$ is the mid - point of $\overline{JK}$, then $JM=MK$. So, $6x + 11=9x−13$.

Step2: Solve the equation for $x$

Subtract $6x$ from both sides: $6x+11−6x=9x−13−6x$, which gives $11 = 3x−13$. Add 13 to both sides: $11 + 13=3x−13 + 13$, so $24 = 3x$. Divide both sides by 3: $\frac{24}{3}=\frac{3x}{3}$, and $x = 8$.

Step3: Find the length of $JM$ and $MK$

Substitute $x = 8$ into the expression for $JM$ (or $MK$). $JM=6x + 11=6\times8+11=48 + 11=59$, $MK=9x−13=9\times8−13=72 - 13=59$.

For the first segment:

Answer:

$x = 3$, $EM=15$, $MF = 15$
For the second segment:
$x = 8$, $JM=59$, $MK=59$