Question

find the value of the variable and yz if y is between x and z.

1. xy = 11, yz = 4c, xz = 83
2. xy = 6b, yz = 8b, xz = 175
3. xy = 7a, yz = 5a, xz = 6a + 24
4. xy = 5.5, yz = 2c, xz = 8.9
5. xy = 5n, yz = 2n, xz = 91
6. xy = 4w, yz = 6w, xz = 12w − 8
7. xy = 11d, yz = 9d − 2, xz = 5d + 28
8. xy = 4n + 3, yz = 2n − 7, xz = 20
9. xy = 3a − 4, yz = 6a + 2, xz = 5a + 22
10. xy = 3k − 2, yz = 7k + 4, xz = 4k + 38
11. xy = 4x, yz = x, and xz = 25
12. xy = 4x, yz = 3x, and xz = 42
13. xy = 12, yz = 2x, and xz = 28
14. xy = 2x + 1, yz = 6x, and xz = 81

line segments

Explanation:

Let's solve problem 10 as an example (we can solve others similarly following the segment addition postulate: if \( Y \) is between \( X \) and \( Z \), then \( XY + YZ = XZ \)).

Problem 10:

Given \( XY = 11 \), \( YZ = 4c \), \( XZ = 83 \)

Step 1: Apply Segment Addition Postulate

Since \( Y \) is between \( X \) and \( Z \), \( XY + YZ = XZ \). Substitute the given values:
\( 11 + 4c = 83 \)

Step 2: Solve for \( c \)

Subtract 11 from both sides:
\( 4c = 83 - 11 \)
\( 4c = 72 \)

Divide both sides by 4:
\( c = \frac{72}{4} \)
\( c = 18 \)

Step 3: Find \( YZ \)

Substitute \( c = 18 \) into \( YZ = 4c \):
\( YZ = 4(18) = 72 \)

Step 1: Apply Segment Addition Postulate

\( XY + YZ = XZ \)
\( 6b + 8b = 175 \)

Step 2: Solve for \( b \)

Combine like terms:
\( 14b = 175 \)

Divide by 14:
\( b = \frac{175}{14} = 12.5 \)

Step 3: Find \( YZ \)

\( YZ = 8b = 8(12.5) = 100 \)

Step 1: Apply Segment Addition Postulate

\( XY + YZ = XZ \)
\( 7a + 5a = 6a + 24 \)

Step 2: Solve for \( a \)

Combine like terms:
\( 12a = 6a + 24 \)

Subtract \( 6a \) from both sides:
\( 6a = 24 \)

Divide by 6:
\( a = 4 \)

Step 3: Find \( YZ \)

\( YZ = 5a = 5(4) = 20 \)

Answer:

\( c = 18 \), \( YZ = 72 \)

Problem 11:

Given \( XY = 6b \), \( YZ = 8b \), \( XZ = 175 \)