Question

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

Explanation:

Step1: Set AB equal to BC (bisect means equal)

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

Step2: Solve for x

Subtract \( 2x \) from both sides: \( 4 - 7x = 25 \)
Subtract 4 from both sides: \( -7x = 21 \)
Divide by -7: \( x = -3 \)

Step3: Find AB and BC

First, find \( AB \): \( AB = 4 - 5(-3) = 4 + 15 = 19 \)
Then, find \( BC \): \( BC = 2(-3) + 25 = -6 + 25 = 19 \) (checks out, since \( AB = BC \))

Step4: Find AC (AB + BC)

\( AC = AB + BC = 19 + 19 = 38 \)

Answer:

\( 38 \)