Question

segment bisectors and midpoints
directions: choose a different column to work on from your partner. put your answers in the provided box. you should both receive the same answer.
question 1
$overline{pm}$ bisects $overline{lh}$ at $m$. if $lm = 3x - 2$ and $mh = 8x - 12$, solve for $x$.
$overline{pm}$ bisects $overline{ln}$ at $m$. if $lm = 10x - 2$ and $mn = 11x - 4$, find the value of $x$.
question 2
$overline{rt}$ bisects $overline{xy}$ at $t$. if $xt = 5x$ and $ty = x + 8$, find $xt$.
$overline{rt}$ bisects $overline{xy}$ at $t$. if $xt = 3x - 2$ and $ty = x + $, find $ty$.
$overline{aj}$ bisects $overline{kp}$ at $m$. if $km = 12 - x$ and $mp = 4 + 3x$, find $kp$.
$overline{aj}$ bisects $overline{kp}$ at $m$. if $km = 18 - x$ and $mp = 2x - 6$, find $pk$.
$overline{ef}$ bisects $overline{gh}$ at $h$. if $ef = 10x - 14$ and $ + 2$, find $hf$.
$overline{ef}$ bisects $overline{gh}$ at $h$. if $eh = 5x - 7$ and $ef = 9x - 11$, find $hf$.

Explanation:

Let's solve the first sub - question: $\overline{PM}$ bisects $\overline{LH}$ at $M$. If $LM = 3x - 2$ and $MH=8x - 12$, solve for $x$.

Step 1: Recall the definition of a segment bisector

A segment bisector divides a segment into two equal parts. So, if $\overline{PM}$ bisects $\overline{LH}$ at $M$, then $LM = MH$.

Step 2: Set up the equation

We know that $LM = 3x - 2$ and $MH = 8x - 12$. Since $LM = MH$, we can set up the equation:
$3x - 2=8x - 12$

Step 3: Solve for $x$

First, subtract $3x$ from both sides of the equation:
$- 2=8x - 3x-12$
Simplify the right - hand side:
$-2 = 5x-12$
Then, add 12 to both sides:
$5x=-2 + 12$
$5x = 10$
Divide both sides by 5:
$x=\frac{10}{5}=2$

Answer:

$x = 2$