vocabulary and core concept check
1. vocabulary if a point, ray, line, line - segment, or plane intersects then what does it do to the segment?
2. complete the sentence to find the length of $overline{ab}$, with endpoints you can use the
monitoring progress and modeling with mathema
in exercises 3 - 6, identify the segment bisector of $overline{rs}$. then find $rs$. (see example 1.)
in exercises 9 $overline{xy}$. then find
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construction in and construct a segr label the midpoint $m$
in exercises 7 and 8, identify the segment bisector of $overline{jk}$. then find $jm$. (see example 2.)
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Accepted Answer

1. **Brief Explanation**: When a point, ray, line, line - segment, or plane intersects a line - segment at its mid - point, it bisects the segment.
   - If it intersects at any other point, it just divides the segment into two sub - segments.
# Answer: If it intersects at the mid - point, it bisects the segment; otherwise, it divides the segment into two sub - segments.

2. **Brief Explanation**: The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ in a coordinate plane is used to find the length of a line - segment with those endpoints.
# Answer: distance formula

3. **Explanation**:
## Step1: Identify the segment bisector
The line $k$ is the segment bisector of $\overline{RS}$ since it intersects $\overline{RS}$ at its mid - point $M$.
## Step2: Find $RS$
Since $RM = 17$ and the bisector divides the segment into two equal parts, $RS=2\times RM$.
$RS = 2\times17=34$
# Answer: The segment bisector is line $k$, $RS = 34$

4. **Explanation**:
## Step1: Identify the segment bisector
The ray $\overrightarrow{AM}$ is the segment bisector of $\overline{RS}$ as it intersects $\overline{RS}$ at its mid - point $M$.
## Step2: Find $RS$
Since $MS = 9$, then $RS = 2\times MS$.
$RS=2\times9 = 18$
# Answer: The segment bisector is ray $\overrightarrow{AM}$, $RS = 18$

5. **Explanation**:
## Step1: Identify the segment bisector
The point $M$ is the segment bisector of $\overline{RS}$ (a point can be a bisector when it is the mid - point of the segment).
## Step2: Find $RS$
Since $MS = 22$, then $RS=2\times MS$.
$RS = 2\times22=44$
# Answer: The segment bisector is point $M$, $RS = 44$

6. **Explanation**:
## Step1: Identify the segment bisector
The line $s$ is the segment bisector of $\overline{RS}$ as it intersects $\overline{RS}$ at its mid - point $M$.
## Step2: Find $RS$
Since $RM = 12$, then $RS=2\times RM$.
$RS=2\times12 = 24$
# Answer: The segment bisector is line $s$, $RS = 24$

7. **Explanation**:
## Step1: Since $M$ is the mid - point of $\overline{JK}$, then $JM=MK$
Set up the equation $7x + 5=8x$.
## Step2: Solve for $x$
Subtract $7x$ from both sides of the equation $7x + 5=8x$, we get $x = 5$.
## Step3: Find $JM$
Substitute $x = 5$ into the expression for $JM$, $JM=7x + 5=7\times5+5=40$
# Answer: The segment bisector is point $M$, $JM = 40$

8. **Explanation**:
## Step1: Since the line $l$ is the segment bisector of $\overline{JK}$, then $JM=MK$
Set up the equation $3x + 15=8x+25$.
## Step2: Solve for $x$
Subtract $3x$ from both sides: $15 = 5x+25$.
Subtract 25 from both sides: $- 10=5x$.
Divide both sides by 5: $x=-2$.
## Step3: Find $JM$
Substitute $x = - 2$ into the expression for $JM$, $JM=3x + 15=3\times(-2)+15=9$
# Answer: The segment bisector is line $l$, $JM = 9$

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