Question

guided practice
vocabulary check

1. in the diagram shown at the right, name the mid - point and a segment bisector of ab.

skill check
m is the midpoint of the segment. find the segment lengths.

1. find rm and ms.
2. find fm and mg.
3. find mq and pq.
4. find ym and yz.

m is the midpoint of jk. find the value of the variable.

1. j x - 14 m 33 k
2. j 16r m 48 k

find the coordinates of the midpoint of pr.

1. p(2,3) r(4,1)
2. p(-2,1) r(1,-3)
3. p(-3,0) r(5,6)

practice and applications
extra practice see p. 677.
recognizing midpoints in exercises 11 - 14, determine whether m is the midpoint of ab. explain your reasoning.

1. a b m
2. a m b
3. a m b
4. a m b
5. visualize ftl sketch a line segment, pq, that is bisected by line l at point r.

Explanation:

Step1: Recall mid - point and segment bisector definitions

The mid - point of a line segment divides the segment into two equal parts. A segment bisector is a line, ray, or segment that divides a line segment into two equal parts.
For the first problem (question 1 in the image):
The mid - point of $\overline{AB}$ is point $C$ since it divides $\overline{AB}$ into two equal parts. The segment bisector of $\overline{AB}$ is line $l$ as it passes through the mid - point $C$ of $\overline{AB}$.

Step2: Solve for segment lengths when $M$ is the mid - point

For question 2:

Since $M$ is the mid - point of $\overline{RS}$ and $RS = 6$, then $RM=\frac{1}{2}RS$ and $MS=\frac{1}{2}RS$. So $RM = 3$ and $MS = 3$.

For question 3:

Since $M$ is the mid - point of $\overline{FG}$ and $FG=50$, then $FM=\frac{1}{2}FG$ and $MG=\frac{1}{2}FG$. So $FM = 25$ and $MG = 25$.

For question 4:

If $PM = 4$ and $M$ is the mid - point of $\overline{PQ}$, then $MQ=PM = 4$ and $PQ=PM + MQ=8$.

For question 5:

If $MZ = 20$ and $M$ is the mid - point of $\overline{YZ}$, then $YM = MZ=20$ and $YZ=YM + MZ = 40$.

Step3: Solve for the variable when $M$ is the mid - point

For question 6:

Since $M$ is the mid - point of $\overline{JK}$, then $JM=MK$. So $x - 14=33$, adding 14 to both sides gives $x=47$.

For question 7:

Since $M$ is the mid - point of $\overline{JK}$, then $JM = MK$. So $16r=48$, dividing both sides by 16 gives $r = 3$.

Step4: Find mid - point coordinates

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

For question 8:

$P(2,3)$ and $R(4,1)$, the mid - point is $(\frac{2 + 4}{2},\frac{3+1}{2})=(3,2)$.

For question 9:

$P(-2,1)$ and $R(1,-3)$, the mid - point is $(\frac{-2 + 1}{2},\frac{1+( - 3)}{2})=(-\frac{1}{2},-1)$.

For question 10:

$P(-3,0)$ and $R(5,6)$, the mid - point is $(\frac{-3 + 5}{2},\frac{0 + 6}{2})=(1,3)$.

Step5: Determine if $M$ is the mid - point

For question 11:

$M$ is not the mid - point of $\overline{AB}$ because $AB$ and $BM$ are not of equal length.

For question 12:

$M$ is the mid - point of $\overline{AB}$ because $AM$ and $MB$ are of equal length.

For question 13:

$M$ is the mid - point of $\overline{AB}$ because the segments on either side of $M$ are congruent.

For question 14:

$M$ is not the mid - point of $\overline{AB}$ because the segments on either side of $M$ are not of equal length.

Step6: Sketch the bisected segment

For question 15: Draw a line segment $\overline{PQ}$. Then draw a line $l$ that intersects $\overline{PQ}$ at a point $R$ such that $PR=RQ$.

Answer:

1. Mid - point: $C$, Segment bisector: line $l$
2. $RM = 3$, $MS = 3$
3. $FM = 25$, $MG = 25$
4. $MQ = 4$, $PQ = 8$
5. $YM = 20$, $YZ = 40$
6. $x = 47$
7. $r = 3$
8. $(3,2)$
9. $(-\frac{1}{2},-1)$
10. $(1,3)$
11. $M$ is not the mid - point of $\overline{AB}$
12. $M$ is the mid - point of $\overline{AB}$
13. $M$ is the mid - point of $\overline{AB}$
14. $M$ is not the mid - point of $\overline{AB}$
15. Sketch as described above.