Measuring Segments - Pangmania - Fill and Sign Printable Template ... - Free Printable

Problem Analysis:

The task involves solving problems related to measuring segments and using given information to find unknown values. Let's break down each part of the problem step by step.

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Part 1: Use the figure on the right to find the length of each segment.

The figure shows a number line with points labeled \( A, B, C, D, E \), and their corresponding coordinates. The coordinates are as follows:
- \( A = -4 \)
- \( B = -2 \)
- \( C = 0 \)
- \( D = 2 \)
- \( E = 4 \)

We need to find the lengths of the following segments:
1. \( AB \)
2. \( BC \)
3. \( CD \)
4. \( DE \)
5. \( AC \)
6. \( CE \)

#### Solution:
The length of a segment between two points \( P \) and \( Q \) on a number line is given by:
\[
PQ = |x_Q - x_P|
\]
where \( x_P \) and \( x_Q \) are the coordinates of points \( P \) and \( Q \), respectively.

1. Length of \( AB \):
\[
AB = |B - A| = |-2 - (-4)| = |-2 + 4| = |2| = 2
\]

2. Length of \( BC \):
\[
BC = |C - B| = |0 - (-2)| = |0 + 2| = |2| = 2
\]

3. Length of \( CD \):
\[
CD = |D - C| = |2 - 0| = |2| = 2
\]

4. Length of \( DE \):
\[
DE = |E - D| = |4 - 2| = |2| = 2
\]

5. Length of \( AC \):
\[
AC = |C - A| = |0 - (-4)| = |0 + 4| = |4| = 4
\]

6. Length of \( CE \):
\[
CE = |E - C| = |4 - 0| = |4| = 4
\]

#### Final Answers for Part 1:
\[
\boxed{2, 2, 2, 2, 4, 4}
\]

---

Part 2: Use the figures at the left to find the missing value.

We are given several scenarios involving segments and their lengths. We need to use the given information to find the missing values.

#### Scenario 1:
Given:
- \( PQ = 7 \)
- \( QR = 3 \)
Find: \( PR \)

#### Solution:
Since \( P, Q, R \) are collinear and \( Q \) is between \( P \) and \( R \):
\[
PR = PQ + QR = 7 + 3 = 10
\]

#### Scenario 2:
Given:
- \( PQ = 20 \)
- \( QR = 22 \)
Find: \( PR \)

#### Solution:
Similarly, since \( P, Q, R \) are collinear and \( Q \) is between \( P \) and \( R \):
\[
PR = PQ + QR = 20 + 22 = 42
\]

#### Scenario 3:
Given:
- \( PQ = 15 \)
- \( PR = 25 \)
Find: \( QR \)

#### Solution:
Since \( P, Q, R \) are collinear and \( Q \) is between \( P \) and \( R \):
\[
QR = PR - PQ = 25 - 15 = 10
\]

#### Scenario 4:
Given:
- \( PR = 18 \)
- \( QR = 7 \)
Find: \( PQ \)

#### Solution:
Since \( P, Q, R \) are collinear and \( Q \) is between \( P \) and \( R \):
\[
PQ = PR - QR = 18 - 7 = 11
\]

#### Scenario 5:
Given:
- \( PQ = 10 \)
- \( QR = 8 \)
Find: \( PR \)

#### Solution:
Since \( P, Q, R \) are collinear and \( Q \) is between \( P \) and \( R \):
\[
PR = PQ + QR = 10 + 8 = 18
\]

#### Final Answers for Part 2:
\[
\boxed{10, 42, 10, 11, 18}
\]

---

Part 3: Use the number line below to tell whether the segments are congruent.

We are given pairs of segments and need to determine if they are congruent (i.e., have the same length).

#### Segment Pair 1: \( GH \) and \( HJ \)
From the number line:
- \( G = -3 \)
- \( H = -1 \)
- \( J = 1 \)

Length of \( GH \):
\[
GH = |H - G| = |-1 - (-3)| = |-1 + 3| = |2| = 2
\]

Length of \( HJ \):
\[
HJ = |J - H| = |1 - (-1)| = |1 + 1| = |2| = 2
\]

Since \( GH = HJ \), the segments are congruent.

#### Segment Pair 2: \( IJ \) and \( JK \)
From the number line:
- \( I = 0 \)
- \( J = 1 \)
- \( K = 3 \)

Length of \( IJ \):
\[
IJ = |J - I| = |1 - 0| = |1| = 1
\]

Length of \( JK \):
\[
JK = |K - J| = |3 - 1| = |2| = 2
\]

Since \( IJ \neq JK \), the segments are not congruent.

#### Segment Pair 3: \( HI \) and \( GJ \)
From the number line:
- \( H = -1 \)
- \( I = 0 \)
- \( G = -3 \)
- \( J = 1 \)

Length of \( HI \):
\[
HI = |I - H| = |0 - (-1)| = |0 + 1| = |1| = 1
\]

Length of \( GJ \):
\[
GJ = |J - G| = |1 - (-3)| = |1 + 3| = |4| = 4
\]

Since \( HI \neq GJ \), the segments are not congruent.

#### Final Answers for Part 3:
\[
\boxed{\text{Yes, No, No}}
\]

---

Part 4: Points A, B, and C are collinear. Given \( AB = 10 \), \( BC = 15 \), and \( AC = 5 \). When can this occur?

#### Analysis:
For points \( A, B, \) and \( C \) to be collinear, one of the points must lie between the other two. However, the given lengths do not satisfy the triangle inequality or the properties of collinear points:
- If \( B \) is between \( A \) and \( C \):
\[
AC = AB + BC \implies 5 = 10 + 15 \quad \text{(False)}
\]
- If \( A \) is between \( B \) and \( C \):
\[
BC = AB + AC \implies 15 = 10 + 5 \quad \text{(True, but contradicts \( AC = 5 \))}
\]
- If \( C \) is between \( A \) and \( B \):
\[
AB = AC + BC \implies 10 = 5 + 15 \quad \text{(False)}
\]

Thus, the given lengths cannot occur for collinear points.

#### Final Answer for Part 4:
\[
\boxed{\text{Never}}
\]

---

Part 5: Use the equations on the left to find the requested value.

We are given:
- \( ST = x + 3 \)
- \( TU = 2x + 3 \)
- \( SU = 6x - 4 \)

We need to find:
1. The value of \( x \)
2. The value of \( TU \)
3. The value of \( ST \)

#### Step 1: Express \( SU \) in terms of \( ST \) and \( TU \).
Since \( S, T, U \) are collinear and \( T \) is between \( S \) and \( U \):
\[
SU = ST + TU
\]
Substitute the given expressions:
\[
6x - 4 = (x + 3) + (2x + 3)
\]

#### Step 2: Solve for \( x \).
Simplify the equation:
\[
6x - 4 = x + 3 + 2x + 3
\]
\[
6x - 4 = 3x + 6
\]
Subtract \( 3x \) from both sides:
\[
3x - 4 = 6
\]
Add 4 to both sides:
\[
3x = 10
\]
Divide by 3:
\[
x = \frac{10}{3}
\]

#### Step 3: Find the value of \( TU \).
Substitute \( x = \frac{10}{3} \) into \( TU = 2x + 3 \):
\[
TU = 2\left(\frac{10}{3}\right) + 3 = \frac{20}{3} + 3 = \frac{20}{3} + \frac{9}{3} = \frac{29}{3}
\]

#### Step 4: Find the value of \( ST \).
Substitute \( x = \frac{10}{3} \) into \( ST = x + 3 \):
\[
ST = \frac{10}{3} + 3 = \frac{10}{3} + \frac{9}{3} = \frac{19}{3}
\]

#### Final Answers for Part 5:
\[
\boxed{\frac{10}{3}, \frac{29}{3}, \frac{19}{3}}
\]

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Final Summary of All Answers:

1. \(\boxed{2, 2, 2, 2, 4, 4}\)
2. \(\boxed{10, 42, 10, 11, 18}\)
3. \(\boxed{\text{Yes, No, No}}\)
4. \(\boxed{\text{Never}}\)
5. \(\boxed{\frac{10}{3}, \frac{29}{3}, \frac{19}{3}}\)