check your understanding
15. explain how to find the distance between two points on a number line.
16. mekhi knows that gh = 7 and hj = 7. based on this information, he claims that point h is the mid - point of $\overline{gj}$. is mekhis claim necessarily true? make a sketch that supports your answer.
17. given: t is the mid - point of $\overline{jk}$, jk = 5x - 3, and jt = 2x + 1. determine the length of $\overline{jk}$.
18. reason quantitatively. $\overline{ab}$ lies on a number line. the coordinate of point a is - 6. given that ab = 20, what are the two possible coordinates for point b?

lesson 4 - 1 practice
19. given: point k is between points h and j, hk = x - 5, kj = 5x - 12, and hj = 25. find the value of x.
20. if b is the mid - point of $\overline{ac}$, ab = x + 6, and ac = 5x - 6, then what is bc?
21. point p is between points f and g. the distance between points f and p is $\frac{1}{4}$ of fg. what is the coordinate of point p?
22. use appropriate tools strategically. anne has a broken ruler. it starts at the 3 - inch mark and ends at the 12 - inch mark. explain how anne could use the ruler to measure the length of a line segment in inches.
23. if p is the mid - point of $\overline{st}$, sp = x + 4, and st = 4x, determine the length of $\overline{st}$.
24. compare and contrast a postulate and a conjecture.

Question

check your understanding
15. explain how to find the distance between two points on a number line.
16. mekhi knows that gh = 7 and hj = 7. based on this information, he claims that point h is the mid - point of $\overline{gj}$. is mekhis claim necessarily true? make a sketch that supports your answer.
17. given: t is the mid - point of $\overline{jk}$, jk = 5x - 3, and jt = 2x + 1. determine the length of $\overline{jk}$.
18. reason quantitatively. $\overline{ab}$ lies on a number line. the coordinate of point a is - 6. given that ab = 20, what are the two possible coordinates for point b?

lesson 4 - 1 practice
19. given: point k is between points h and j, hk = x - 5, kj = 5x - 12, and hj = 25. find the value of x.
20. if b is the mid - point of $\overline{ac}$, ab = x + 6, and ac = 5x - 6, then what is bc?
21. point p is between points f and g. the distance between points f and p is $\frac{1}{4}$ of fg. what is the coordinate of point p?
22. use appropriate tools strategically. anne has a broken ruler. it starts at the 3 - inch mark and ends at the 12 - inch mark. explain how anne could use the ruler to measure the length of a line segment in inches.
23. if p is the mid - point of $\overline{st}$, sp = x + 4, and st = 4x, determine the length of $\overline{st}$.
24. compare and contrast a postulate and a conjecture.

Accepted Answer

### 15.
# Explanation:
## Step1: Identify coordinates
Let the two points on the number - line have coordinates $a$ and $b$.
## Step2: Calculate distance
The distance $d$ between them is $d = |a - b|$. This is because distance is non - negative, and the absolute - value function ensures that we get a positive result regardless of whether $a>b$ or $a < b$.
# Answer:
The distance between two points on a number line with coordinates $a$ and $b$ is $|a - b|$.

### 16.
# Explanation:
Mekhi's claim is not necessarily true. Just because $GH = 7$ and $HJ=7$ does not mean that $G$, $H$, and $J$ are collinear. If $G$, $H$, and $J$ form a triangle (non - collinear points), then $H$ is not the mid - point of $\overline{GJ}$. To support this, we can draw a triangle $\triangle GHJ$ where $GH = 7$ and $HJ = 7$.
# Answer:
No. A sketch of a non - collinear triangle $\triangle GHJ$ with $GH = 7$ and $HJ = 7$ can be drawn to show that $H$ is not the mid - point of $\overline{GJ}$.

### 17.
# Explanation:
## Step1: Use mid - point property
Since $T$ is the mid - point of $\overline{JK}$, then $JT=\frac{1}{2}JK$.
## Step2: Substitute given expressions
We know that $JK = 5x−3$ and $JT = 2x + 1$. So, $2(2x + 1)=5x−3$.
## Step3: Solve the equation
Expand the left - hand side: $4x+2 = 5x−3$. Subtract $4x$ from both sides: $2=x - 3$. Add 3 to both sides: $x = 5$.
## Step4: Find the length of $JK$
Substitute $x = 5$ into the expression for $JK$: $JK=5x−3=5\times5−3=25 - 3=22$.
# Answer:
22

### 18.
# Explanation:
## Step1: Set up the absolute - value equation
Let the coordinate of point $B$ be $x$. The distance between $A$ and $B$ is $|x-(-6)|=|x + 6|$. Since $AB = 20$, we have the equation $|x + 6|=20$.
## Step2: Solve the absolute - value equation
Case 1: $x+6 = 20$, then $x=20 - 6=14$. Case 2: $x + 6=-20$, then $x=-20 - 6=-26$.
# Answer:
14 and $-26$

### 19.
# Explanation:
## Step1: Use the segment addition postulate
Since $K$ is between $H$ and $J$, $HK+KJ=HJ$.
## Step2: Substitute given expressions
We have $(x - 5)+(5x-12)=25$.
## Step3: Simplify the left - hand side
Combine like terms: $x-5 + 5x-12=6x-17$. So, $6x-17 = 25$.
## Step4: Solve for $x$
Add 17 to both sides: $6x=25 + 17=42$. Divide both sides by 6: $x = 7$.
# Answer:
7

### 20.
# Explanation:
## Step1: Use mid - point property
Since $B$ is the mid - point of $\overline{AC}$, $AB=\frac{1}{2}AC$.
## Step2: Substitute given expressions
We know that $AB=x + 6$ and $AC = 5x−6$. So, $2(x + 6)=5x−6$.
## Step3: Solve the equation
Expand the left - hand side: $2x+12 = 5x−6$. Subtract $2x$ from both sides: $12=3x−6$. Add 6 to both sides: $18 = 3x$. Divide both sides by 3: $x = 6$.
## Step4: Find $BC$
Since $AB = BC$ (because $B$ is the mid - point of $\overline{AC}$), and $AB=x + 6$, substituting $x = 6$ gives $AB=6 + 6=12$. So, $BC = 12$.
# Answer:
12

### 21.
# Explanation:
## Step1: Determine the distance between $F$ and $G$
The distance between $F$ and $G$ is $22 - 4=18$.
## Step2: Find the distance between $F$ and $P$
Since the distance between $F$ and $P$ is $\frac{1}{4}$ of $FG$, then $FP=\frac{1}{4}\times18 = 4.5$.
## Step3: Calculate the coordinate of $P$
The coordinate of $F$ is 4. So the coordinate of $P$ is $4+4.5 = 8.5$.
# Answer:
8.5

### 22.
# Explanation:
Anne can place the 3 - inch mark at one end of the line segment. Then, she can mark the point on the line segment that aligns with the 12 - inch mark on the ruler. The length of the line segment $L$ can be found by subtracting 3 from the reading on the ruler at the other end of the line segment. If the reading on the ruler at the other end of the line segment is $r$ (where $3\leq r\leq12$), then $L=r - 3$.
# Answer:
Place the 3 - inch mark at one end of the line segment. Mark the point on the line segment that aligns with the 12 - inch mark on the ruler. The length of the line segment is the reading on the ruler at the other end minus 3.

### 23.
# Explanation:
## Step1: Use mid - point property
Since $P$ is the mid - point of $\overline{ST}$, then $SP=\frac{1}{2}ST$.
## Step2: Substitute given expressions
We know that $SP=x + 4$ and $ST = 4x$. So, $x + 4=\frac{1}{2}(4x)$.
## Step3: Solve the equation
Simplify the right - hand side: $x + 4 = 2x$. Subtract $x$ from both sides: $4=x$.
## Step4: Find the length of $ST$
Substitute $x = 4$ into the expression for $ST$: $ST = 4x=4\times4 = 16$.
# Answer:
16

### 24.
# Explanation:
A postulate is a statement that is accepted as true without proof. It is a fundamental building block in a mathematical system. A conjecture is an unproven statement that is based on observations or patterns. While a postulate is assumed to be true from the start, a conjecture needs to be proven or disproven through logical reasoning and mathematical methods.
# Answer:
A postulate is an accepted - without - proof statement, while a conjecture is an unproven statement based on observations.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

15.

Step1: Identify coordinates

Step2: Calculate distance

Step1: Use mid - point property

Step2: Substitute given expressions

Step3: Solve the equation

Step4: Find the length of \(JK\)

Answer:

16.