test chapter 1
part b:
1. if ln = 6x - 5, lm = x + 7, and mn = 3x + 20, find x and mn.
2. given the graph below, find mn. round to the nearest hundredths.
3. given the points below, find xy. round to the nearest hundredths.
x(2, 4) and y(6, 3)
4. find the midpoint of ab if a(-3, 8) and b(-7, -6)
5. if m is the midpoint of ab, find the coordinates of b if a(11, -2) and m(4, 3)
6. if line n bisects ce, find x and cd.
7. find the value of x.
8. find the value of x.

Question

Accepted Answer

1.
# Explanation:
## Step1: Use segment - addition postulate
Since $LN = LM+MN$, we substitute the given expressions: $6x - 5=(x + 7)+(3x + 20)$.
## Step2: Simplify the right - hand side
$6x-5=x + 7+3x + 20$, which simplifies to $6x-5 = 4x+27$.
## Step3: Solve for x
Subtract $4x$ from both sides: $6x-4x-5=4x - 4x+27$, so $2x-5 = 27$. Then add 5 to both sides: $2x-5 + 5=27 + 5$, giving $2x=32$. Divide both sides by 2: $x = 16$.
## Step4: Find MN
Substitute $x = 16$ into the expression for $MN$: $MN=3x + 20=3\times16+20=48 + 20=68$.
# Answer:
$x = 16$, $MN = 68$

3.
# Explanation:
## Step1: Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$
Here, $x_1 = 2,y_1 = 4,x_2 = 6,y_2 = 3$.
## Step2: Substitute the values
$XY=\sqrt{(6 - 2)^2+(3 - 4)^2}=\sqrt{4^2+( - 1)^2}=\sqrt{16 + 1}=\sqrt{17}\approx4.12$.
# Answer:
$XY\approx4.12$

4.
# Explanation:
## Step1: Use the mid - point formula $(\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})$
Here, $x_1=-3,y_1 = 8,x_2=-7,y_2=-6$.
## Step2: Calculate the coordinates
$\frac{-3+( - 7)}{2}=\frac{-3-7}{2}=\frac{-10}{2}=-5$ and $\frac{8+( - 6)}{2}=\frac{8 - 6}{2}=1$.
# Answer:
$(-5,1)$

5.
# Explanation:
## Step1: Use the mid - point formula $(\frac{x_A+x_B}{2},\frac{y_A + y_B}{2})=(x_M,y_M)$
We know $x_A = 11,y_A=-2,x_M = 4,y_M = 3$. For the x - coordinate: $\frac{11+x_B}{2}=4$, multiply both sides by 2: $11+x_B = 8$, then $x_B=8 - 11=-3$. For the y - coordinate: $\frac{-2+y_B}{2}=3$, multiply both sides by 2: $-2+y_B = 6$, then $y_B=6 + 2 = 8$.
# Answer:
$(-3,8)$

6.
# Explanation:
## Step1: Since line $n$ bisects $CE$, then $CD=DE$
So $x + 6=4x-21$.
## Step2: Solve for x
Subtract $x$ from both sides: $x-x + 6=4x-x-21$, so $6 = 3x-21$. Add 21 to both sides: $6+21=3x-21 + 21$, giving $27 = 3x$. Divide both sides by 3: $x = 9$.
## Step3: Find CD
Substitute $x = 9$ into the expression for $CD$: $CD=x + 6=9+6=15$.
# Answer:
$x = 9$, $CD = 15$

7.
# Explanation:
## Step1: Since the angles are vertical angles, they are equal
$7x-1=6x - 1$.
## Step2: Solve for x
Subtract $6x$ from both sides: $7x-6x-1=6x-6x - 1$, so $x-1=-1$. Add 1 to both sides: $x-1 + 1=-1 + 1$, giving $x = 0$.
# Answer:
$x = 0$

8.
# Explanation:
## Step1: Since the angles are vertical angles, they are equal
$5x + 4=8x-71$.
## Step2: Solve for x
Subtract $5x$ from both sides: $5x-5x + 4=8x-5x-71$, so $4 = 3x-71$. Add 71 to both sides: $4+71=3x-71 + 71$, giving $75 = 3x$. Divide both sides by 3: $x = 25$.
# Answer:
$x = 25$

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QUESTION IMAGE

Question

Explanation:

Step1: Use segment - addition postulate

Step2: Simplify the right - hand side

Step3: Solve for x

Step4: Find MN

Step1: Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Step2: Substitute the values

Step1: Use the mid - point formula \((\frac{x_1+x_2}{2},\frac{y_1 + y_2}{2})\)

Step2: Calculate the coordinates

Answer: