Question

1. find the coordinates of a if m(-1, 2) is the midpoint of $overline{ab}$ and b has coordinates of (3, -5).
2. find the coordinates of j if k(-5, 10) is the midpoint of $overline{jl}$ and l has coordinates of (-8, 6).
3. find the coordinates of r if q(-1, 3) is the midpoint of $overline{pr}$ and p has coordinates of (5, 6).
4. if p is the midpoint of $overline{xy}$, $xp = 8x - 2$, and $py = 12x - 30$, find the value of x.
5. if g is the midpoint of $overline{fh}$, $fg = 14x + 25$, and $gh = 73 - 2x$, find fh.
6. using the diagram to the left, if line n bisects $overline{qr}$, find qp

Explanation:

Step1: Recall mid - point formula

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

Step2: Solve for question 6

Let the coordinates of $A$ be $(x,y)$. We know $M(-1,2)$ is the mid - point of $AB$ and $B(3,-5)$. Using the mid - point formula:
For the $x$ - coordinate: $\frac{x + 3}{2}=-1$. Cross - multiply gives $x + 3=-2$, so $x=-2 - 3=-5$.
For the $y$ - coordinate: $\frac{y+( - 5)}{2}=2$. Cross - multiply gives $y-5 = 4$, so $y=4 + 5=9$. The coordinates of $A$ are $(-5,9)$.

Step3: Solve for question 7

Let the coordinates of $J$ be $(x,y)$. We know $K(-5,10)$ is the mid - point of $JL$ and $L(-8,6)$. Using the mid - point formula:
For the $x$ - coordinate: $\frac{x+( - 8)}{2}=-5$. Cross - multiply gives $x-8=-10$, so $x=-10 + 8=-2$.
For the $y$ - coordinate: $\frac{y + 6}{2}=10$. Cross - multiply gives $y+6 = 20$, so $y=20 - 6=14$. The coordinates of $J$ are $(-2,14)$.

Step4: Solve for question 8

Let the coordinates of $R$ be $(x,y)$. We know $Q(-1,3)$ is the mid - point of $PR$ and $P(5,6)$. Using the mid - point formula:
For the $x$ - coordinate: $\frac{x + 5}{2}=-1$. Cross - multiply gives $x+5=-2$, so $x=-2 - 5=-7$.
For the $y$ - coordinate: $\frac{y + 6}{2}=3$. Cross - multiply gives $y+6 = 6$, so $y=6 - 6=0$. The coordinates of $R$ are $(-7,0)$.

Step5: Solve for question 9

Since $P$ is the mid - point of $\overline{XY}$, then $XP = PY$. So $8x-2=12x-30$.
Subtract $8x$ from both sides: $-2=12x-8x-30$, which simplifies to $-2 = 4x-30$.
Add 30 to both sides: $4x=-2 + 30=28$.
Divide both sides by 4: $x = 7$.

Step6: Solve for question 10

Since $G$ is the mid - point of $\overline{FH}$, then $FG=GH$. So $14x + 25=73-2x$.
Add $2x$ to both sides: $14x+2x+25=73$, which simplifies to $16x+25=73$.
Subtract 25 from both sides: $16x=73 - 25=48$.
Divide both sides by 16: $x = 3$.
$FG=14x + 25=14\times3+25=42 + 25=67$.
$FH=FG + GH$, and since $FG = GH$, $FH=2FG=2\times67 = 134$.

Answer:

1. $(-5,9)$
2. $(-2,14)$
3. $(-7,0)$
4. $7$
5. $134$