Question

find the indicated value (3 points each):

1. the graph below shows a bridge between two islands. if point m is the mid - point of $overline{pq}$, and $overline{pm}=6cm$, what is the length of $overline{pq}$?
2. $kn$ is the median of $\triangle klm$, find the coordinates of point m.
3. the endpoints of a line segment are (4, s) and (w, - 1). the mid - point of this line segment is (-4, 12). solve for s and w.
4. the endpoints of a line segment are (9, - 5) and (-7, 11). the mid - point of this line segment is (c, 3). solve for c.

Explanation:

Step1: Recall mid - point property for 13

If M is the mid - point of $\overline{PQ}$, then $PM = MQ$. Given $PM = 6$ cm, so $PQ=PM + MQ$.
Since $PM = MQ = 6$ cm, then $PQ=6 + 6$.
$PQ = 12$ cm

Step2: Recall mid - point formula for 14

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. In $\triangle KLM$, since $KN$ is the median, $N$ is the mid - point of $\overline{LM}$. Let $L(-9,5)$ and $M(x,y)$. The x - coordinate of the mid - point $N(-7,5)$ of $\overline{LM}$: $\frac{-9 + x}{2}=-7$. Solving for $x$ gives $-9 + x=-14$, so $x=-5$. The y - coordinate of the mid - point of $\overline{LM}$: $\frac{5 + y}{2}=5$, so $5 + y = 10$ and $y = 5$. So the coordinates of $M$ are $(-5,5)$.

Step3: Use mid - point formula for 15

The mid - point formula for two points $(x_1,y_1)=(4,s)$ and $(x_2,y_2)=(w,-1)$ with mid - point $(x_m,y_m)=(-4,12)$ gives two equations. For the x - coordinate: $\frac{4 + w}{2}=-4$, so $4+w=-8$ and $w=-12$. For the y - coordinate: $\frac{s+( - 1)}{2}=12$, so $s - 1 = 24$ and $s = 25$.

Step4: Use mid - point formula for 16

The mid - point formula for two points $(x_1,y_1)=(9,-5)$ and $(x_2,y_2)=(-7,11)$ with mid - point $(x_m,y_m)=(c,3)$. For the x - coordinate: $\frac{9+( - 7)}{2}=c$. Then $\frac{9 - 7}{2}=c$, so $c = 1$.

Answer:

1. $12$ cm
2. $(-5,5)$
3. $s = 25$, $w=-12$
4. $c = 1$