Question

1. if line y bisects (overline{ac}), (ab = 4 - 5x), and (bc = 2x + 25), find (ac).

Explanation:

Step1: Use the property of a bisected segment

Since line \( y \) bisects \( \overline{AC} \), then \( AB = BC \). So we set up the equation \( 4-5x=2x + 25 \).

Step2: Solve the equation for \( x \)

Add \( 5x \) to both sides: \( 4=2x+25 + 5x \), which simplifies to \( 4=7x + 25 \).
Subtract \( 25 \) from both sides: \( 4-25=7x \), so \( - 21=7x \).
Divide both sides by \( 7 \): \( x=\frac{-21}{7}=-3 \).

Step3: Find the length of \( AB \) or \( BC \)

Substitute \( x =- 3 \) into \( AB = 4-5x \). Then \( AB=4-5\times(-3)=4 + 15=19 \). (We could also substitute into \( BC=2x + 25 \): \( BC=2\times(-3)+25=-6 + 25=19 \))

Step4: Calculate \( AC \)

Since \( AC=AB + BC \) and \( AB = BC = 19 \), then \( AC=19+19=38 \).

Answer:

\( 38 \)