Question

ii. write and solve an equation to find the length of each segment.

1. b is between a and c. ab = x, bc = 2x, ac = 18. find ab and bc.
2. b is between e and d. bd = t + 9, eb = t, ed = 31. find bd and eb

Explanation:

Problem 7

Step1: Use segment addition postulate

Since \( B \) is between \( A \) and \( C \), we have \( AB + BC = AC \). Substituting the given values \( AB = x \), \( BC = 2x \), and \( AC = 18 \), we get the equation \( x + 2x = 18 \).

Step2: Solve the equation for \( x \)

Combine like terms: \( 3x = 18 \). Then divide both sides by 3: \( x = \frac{18}{3} = 6 \).

Step3: Find \( AB \) and \( BC \)

Since \( AB = x \), \( AB = 6 \). Since \( BC = 2x \), substitute \( x = 6 \): \( BC = 2 \times 6 = 12 \).

Step1: Use segment addition postulate

Since \( B \) is between \( E \) and \( D \), we have \( EB + BD = ED \). Substituting the given values \( EB = t \), \( BD = t + 9 \), and \( ED = 31 \), we get the equation \( t + (t + 9) = 31 \).

Step2: Solve the equation for \( t \)

Combine like terms: \( 2t + 9 = 31 \). Subtract 9 from both sides: \( 2t = 31 - 9 = 22 \). Then divide both sides by 2: \( t = \frac{22}{2} = 11 \).

Step3: Find \( BD \) and \( EB \)

Since \( EB = t \), \( EB = 11 \). Since \( BD = t + 9 \), substitute \( t = 11 \): \( BD = 11 + 9 = 20 \).

Answer:

\( AB = 6 \), \( BC = 12 \)

Problem 8