Question

1. if bd = 7x - 10, bc = 4x - 29, and cd = 5x - 9, find each value.

Explanation:

Assuming the points are colinear with \( B \), \( C \), \( D \) such that \( BD = BC + CD \) (segment addition postulate).

Step 1: Set up the equation

Since \( BD = BC + CD \), substitute the given expressions:
\( 7x - 10=(4x - 29)+(5x - 9) \)

Step 2: Simplify the right - hand side

Combine like terms on the right:
\( 4x+5x=9x \) and \( - 29-9=-38 \), so the equation becomes \( 7x - 10 = 9x-38 \)

Step 3: Solve for \( x \)

Subtract \( 7x \) from both sides:
\( 7x-7x - 10=9x - 7x-38 \)
\( - 10 = 2x-38 \)

Add 38 to both sides:
\( - 10 + 38=2x-38 + 38 \)
\( 28 = 2x \)

Divide both sides by 2:
\( x=\frac{28}{2}=14 \)

Step 4: Find \( BC \)

Substitute \( x = 14 \) into \( BC = 4x-29 \):
\( BC=4\times14 - 29=56 - 29 = 27 \)

Step 5: Find \( CD \)

Substitute \( x = 14 \) into \( CD = 5x-9 \):
\( CD=5\times14-9 = 70 - 9=61 \)

Step 6: Find \( BD \)

Substitute \( x = 14 \) into \( BD = 7x-10 \):
\( BD=7\times14-10 = 98 - 10 = 88 \)
We can also check using \( BD=BC + CD=27 + 61 = 88 \)

Answer:

\( x = 14 \), \( BC=27 \), \( CD = 61 \), \( BD = 88 \)