examples 5 and 6
suppose m is the mid - point of (overline{fg}). find each missing measure.
39. (fm = 5y + 13, mg = 5 - 3y, fg=?)
40. (fm = 3x - 4, mg = 5x - 26, fg =?)
41. (fm = 8a + 1, fg = 42, a =?)
42. (mg = 7x - 15, fg = 33, x =?)
43. (fm = 3n + 1, mg = 6 - 2n, fg =?)
44. (fm = 12x - 4, mg = 5x + 10, fg =?)
45. (fm = 2k - 5, fg = 18, k =?)
46. (fg = 14a + 1, fm = 14.5, a =?)
47. (mg = 13x + 1, fg = 15, x =?)
48. (fg = 11x - 15.6, mg = 10.9, x =?)
mixed exercises
find the coordinates of the missing endpoint if p is the mid - point of (overline{nq}).
49. (n(2,0),p(5,2))
50. (n(5,4),p(6,3))
51. (q(3,9),p(-1,5))

Question

Accepted Answer

### 39.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, then $FM = MG$. So, $5y + 13=5 - 3y$.
## Step2: Solve for $y$
Add $3y$ to both sides: $5y+3y + 13=5-3y + 3y$, which gives $8y+13 = 5$. Then subtract 13 from both sides: $8y+13 - 13=5 - 13$, so $8y=-8$. Divide both sides by 8: $y=-1$.
## Step3: Find $FM$ and $MG$
Substitute $y = - 1$ into $FM$: $FM=5(-1)+13=8$. Substitute $y=-1$ into $MG$: $MG = 5-3(-1)=8$.
## Step4: Find $FG$
Since $FG=FM + MG$, then $FG=8 + 8=16$.
# Answer:
$FG = 16$

### 40.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FM = MG$. So, $3x - 4=5x - 26$.
## Step2: Solve for $x$
Subtract $3x$ from both sides: $3x-3x - 4=5x-3x - 26$, which gives $-4 = 2x-26$. Add 26 to both sides: $-4 + 26=2x-26 + 26$, so $22 = 2x$. Divide both sides by 2: $x = 11$.
## Step3: Find $FM$ and $MG$
Substitute $x = 11$ into $FM$: $FM=3(11)-4=29$. Substitute $x = 11$ into $MG$: $MG=5(11)-26=29$.
## Step4: Find $FG$
Since $FG=FM + MG$, then $FG=29+29 = 58$.
# Answer:
$FG = 58$

### 41.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FG = 2FM$. Given $FM=8a + 1$ and $FG = 42$, we have $42=2(8a + 1)$.
## Step2: Solve the equation
First, distribute on the right - hand side: $42=16a+2$. Subtract 2 from both sides: $42 - 2=16a+2 - 2$, so $40 = 16a$. Divide both sides by 16: $a=\frac{40}{16}=\frac{5}{2}=2.5$.
# Answer:
$a = 2.5$

### 42.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FG = 2MG$. Given $MG=7x - 15$ and $FG = 33$, we have $33=2(7x - 15)$.
## Step2: Solve the equation
First, distribute on the right - hand side: $33=14x-30$. Add 30 to both sides: $33 + 30=14x-30 + 30$, so $63 = 14x$. Divide both sides by 14: $x=\frac{63}{14}=4.5$.
# Answer:
$x = 4.5$

### 43.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FM = MG$. So, $3n + 1=6 - 2n$.
## Step2: Solve for $n$
Add $2n$ to both sides: $3n+2n + 1=6-2n + 2n$, which gives $5n+1 = 6$. Subtract 1 from both sides: $5n+1 - 1=6 - 1$, so $5n = 5$. Divide both sides by 5: $n = 1$.
## Step3: Find $FM$ and $MG$
Substitute $n = 1$ into $FM$: $FM=3(1)+1=4$. Substitute $n = 1$ into $MG$: $MG=6-2(1)=4$.
## Step4: Find $FG$
Since $FG=FM + MG$, then $FG=4 + 4=8$.
# Answer:
$FG = 8$

### 44.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FM = MG$. So, $12x - 4=5x + 10$.
## Step2: Solve for $x$
Subtract $5x$ from both sides: $12x-5x - 4=5x-5x + 10$, which gives $7x-4 = 10$. Add 4 to both sides: $7x-4 + 4=10 + 4$, so $7x = 14$. Divide both sides by 7: $x = 2$.
## Step3: Find $FM$ and $MG$
Substitute $x = 2$ into $FM$: $FM=12(2)-4=20$. Substitute $x = 2$ into $MG$: $MG=5(2)+10=20$.
## Step4: Find $FG$
Since $FG=FM + MG$, then $FG=20+20 = 40$.
# Answer:
$FG = 40$

### 45.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FG = 2FM$. Given $FM=2k - 5$ and $FG = 18$, we have $18=2(2k - 5)$.
## Step2: Solve the equation
First, distribute on the right - hand side: $18=4k-10$. Add 10 to both sides: $18 + 10=4k-10 + 10$, so $28 = 4k$. Divide both sides by 4: $k = 7$.
# Answer:
$k = 7$

### 46.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FG = 2FM$. Given $FG=14a + 1$ and $FM = 14.5$, we have $14a+1=2\times14.5$.
## Step2: Solve the equation
First, simplify the right - hand side: $14a+1 = 29$. Subtract 1 from both sides: $14a+1 - 1=29 - 1$, so $14a = 28$. Divide both sides by 14: $a = 2$.
# Answer:
$a = 2$

### 47.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FG = 2MG$. Given $MG=13x + 1$ and $FG = 15$, we have $15=2(13x + 1)$.
## Step2: Solve the equation
First, distribute on the right - hand side: $15=26x+2$. Subtract 2 from both sides: $15 - 2=26x+2 - 2$, so $13 = 26x$. Divide both sides by 26: $x=\frac{13}{26}=0.5$.
# Answer:
$x = 0.5$

### 48.
# Explanation:
## Step1: Use mid - point property
Since $M$ is the mid - point of $\overline{FG}$, $FG = 2MG$. Given $FG=11x - 15.6$ and $MG = 10.9$, we have $11x-15.6=2\times10.9$.
## Step2: Solve the equation
First, simplify the right - hand side: $11x-15.6 = 21.8$. Add 15.6 to both sides: $11x-15.6 + 15.6=21.8 + 15.6$, so $11x = 37.4$. Divide both sides by 11: $x = 3.4$.
# Answer:
$x = 3.4$

### 49.
# Explanation:
## Step1: Use mid - point formula
The mid - point formula is $P(\frac{x_N+x_Q}{2},\frac{y_N + y_Q}{2})$. Let $Q(x,y)$. We have $\frac{2 + x}{2}=5$ and $\frac{0 + y}{2}=2$.
## Step2: Solve for $x$ and $y$
For $\frac{2 + x}{2}=5$, multiply both sides by 2: $2+x = 10$, then $x = 8$. For $\frac{0 + y}{2}=2$, multiply both sides by 2: $y = 4$.
# Answer:
$Q(8,4)$

### 50.
# Explanation:
## Step1: Use mid - point formula
Let $Q(x,y)$. We have $\frac{5 + x}{2}=6$ and $\frac{4 + y}{2}=3$.
## Step2: Solve for $x$ and $y$
For $\frac{5 + x}{2}=6$, multiply both sides by 2: $5+x = 12$, then $x = 7$. For $\frac{4 + y}{2}=3$, multiply both sides by 2: $4 + y=6$, then $y = 2$.
# Answer:
$Q(7,2)$

### 51.
# Explanation:
## Step1: Use mid - point formula
Let $N(x,y)$. We have $\frac{x + 3}{2}=-1$ and $\frac{y + 9}{2}=5$.
## Step2: Solve for $x$ and $y$
For $\frac{x + 3}{2}=-1$, multiply both sides by 2: $x+3=-2$, then $x=-5$. For $\frac{y + 9}{2}=5$, multiply both sides by 2: $y + 9 = 10$, then $y = 1$.
# Answer:
$N(-5,1)$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

39.

Step1: Use mid - point property

Step2: Solve for \(y\)

Step3: Find \(FM\) and \(MG\)

Step4: Find \(FG\)

Step1: Use mid - point property

Step2: Solve for \(x\)

Step3: Find \(FM\) and \(MG\)

Step4: Find \(FG\)

Step1: Use mid - point property

Step2: Solve the equation

Answer:

40.