5. open response find the value of x if q is between p and r, pq = 5x - 10, qr = 3(x + 4), and $\overline{pq}\cong\overline{qr}$. (lesson 1 - 3)
6. open response on a straight highway, the distance from lorettas house to a park is 43 miles. her friend jamal lives along this same highway between lorettas house and the park. the distance from lorettas house to jamals house is 31 miles. how many miles is it from jamals house to the park? (lesson 1 - 3)
7. multiple choice find the distance between the two points on a coordinate plane. (lesson 1 - 4) a(5, 1) and b(-3, -3) a) 4$\sqrt{5}$ b) 4$\sqrt{3}$ c) 2$\sqrt{2}$ d) 2$\sqrt{3}$
8. open response true or false: $\overline{xy}\cong\overline{wz}$ (lesson 1 - 4)
9. multiple choice the coordinates of a and b on a number line are -7 and 9. the coordinates of c and d on a number line are -4 and 12. are $\overline{ab}$ and $\overline{cd}$ congruent? if yes, what is the length of each segment? (lesson 1 - 4) a) no b) yes; 16 c) yes; -16 d) yes; 8
10. open response the coordinate of point x on $\overline{pq}$ that is $\frac{3}{4}$ of the distance from p to q is ___. (lesson 1 - 5)
11. multiple choice on a number line, point s is located at -3 and point t is located at 9. where is point r located on $\overline{st}$ if the ratio of sr to rt is 3:4? (lesson 1 - 5) a) $\frac{27}{7}$ b) 2$\frac{1}{4}$ c) 1$\frac{1}{4}$ d) $\frac{15}{7}$

Question

Accepted Answer

### 5.
# Explanation:
## Step1: Set up the equation
Since $\overline{PQ}\cong\overline{QR}$, we have $5x - 10=3(x + 4)$.
## Step2: Expand the right - hand side
Using the distributive property, $3(x + 4)=3x+12$. So the equation becomes $5x - 10 = 3x+12$.
## Step3: Move the $x$ terms to one side
Subtract $3x$ from both sides: $5x-3x - 10=3x - 3x+12$, which simplifies to $2x-10 = 12$.
## Step4: Move the constant terms to one side
Add 10 to both sides: $2x-10 + 10=12 + 10$, getting $2x=22$.
## Step5: Solve for $x$
Divide both sides by 2: $x = 11$.
# Answer:
$x = 11$

### 6.
# Explanation:
The total distance from Loretta's house to the park is 43 miles and the distance from Loretta's house to Jamal's house is 31 miles. To find the distance from Jamal's house to the park, we subtract the distance from Loretta's house to Jamal's house from the total distance from Loretta's house to the park.
## Step1: Subtract the distances
$43-31 = 12$ miles.
# Answer:
12 miles

### 7.
# Explanation:
The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $x_1 = 5,y_1 = 1,x_2=-3,y_2=-3$.
## Step1: Calculate $(x_2 - x_1)$ and $(y_2 - y_1)$
$x_2 - x_1=-3 - 5=-8$ and $y_2 - y_1=-3 - 1=-4$.
## Step2: Calculate $(x_2 - x_1)^2+(y_2 - y_1)^2$
$(-8)^2+(-4)^2=64 + 16=80$.
## Step3: Calculate the distance
$d=\sqrt{80}=\sqrt{16	imes5}=4\sqrt{5}$.
# Answer:
A. $4\sqrt{5}$

### 8.
# Explanation:
The length of $\overline{XY}=7-(-1)=8$ and the length of $\overline{WZ}=4-(-6)=10$. Since $8
eq10$, $\overline{XY}
ot\cong\overline{WZ}$.
# Answer:
False

### 9.
# Explanation:
The length of a line - segment with endpoints $a$ and $b$ on a number line is $|b - a|$.
For $\overline{AB}$ with $A=-7$ and $B = 9$, the length of $\overline{AB}=|9-(-7)|=|9 + 7|=16$.
For $\overline{CD}$ with $C=-4$ and $D = 12$, the length of $\overline{CD}=|12-(-4)|=|12 + 4|=16$. Since the lengths are equal, $\overline{AB}$ and $\overline{CD}$ are congruent and the length of each segment is 16.
# Answer:
B. yes; 16

### 10.
# Explanation:
The distance from $P=-5$ to $Q = 7$ is $|7-(-5)|=12$. The distance from $P$ to $X$ which is $\frac{3}{4}$ of the distance from $P$ to $Q$ is $\frac{3}{4}	imes12 = 9$. Starting from $P=-5$, moving 9 units to the right gives $-5+9 = 4$.
# Answer:
4

### 11.
# Explanation:
If a point $R$ divides the line - segment $\overline{ST}$ where $S=-3$ and $T = 9$ in the ratio $m:n=3:4$, the coordinate of $R$ is given by the formula $R=\frac{mT+nS}{m + n}$.
## Step1: Substitute the values
$R=\frac{3	imes9+4	imes(-3)}{3 + 4}=\frac{27-12}{7}=\frac{15}{7}$.
# Answer:
D. $\frac{15}{7}$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

5.

Step1: Set up the equation

Step2: Expand the right - hand side

Step3: Move the $x$ terms to one side

Step4: Move the constant terms to one side

Step5: Solve for $x$

Step1: Subtract the distances

Step1: Calculate $(x_2 - x_1)$ and $(y_2 - y_1)$

Step2: Calculate $(x_2 - x_1)^2+(y_2 - y_1)^2$

Step3: Calculate the distance

Answer:

6.