Question

1. points x, y, and z are collinear. point y bisects xz. find xz if xy = 2x - 4; yz = x - 12

Explanation:

Step1: Use the definition of a bisector

Since \( Y \) bisects \( XZ \), we know that \( XY = YZ \). So we set up the equation:
\( 2x - 4 = x - 12 \)

Step2: Solve for \( x \)

Subtract \( x \) from both sides:
\( 2x - x - 4 = x - x - 12 \)
\( x - 4 = -12 \)

Then add 4 to both sides:
\( x - 4 + 4 = -12 + 4 \)
\( x = -8 \)

Step3: Find the length of \( XY \) and \( YZ \)

First, find \( XY \) by substituting \( x = -8 \) into \( XY = 2x - 4 \):
\( XY = 2(-8) - 4 = -16 - 4 = -20 \) (But length can't be negative, wait, maybe I made a mistake? Wait, no, maybe the problem has a typo? Wait, no, let's check again. Wait, maybe the equations are \( XY = 2x + 4 \) and \( YZ = x - 12 \)? Wait, the original problem says \( XY = 2x - 4 \), \( YZ = x - 12 \). Wait, but if \( Y \) bisects \( XZ \), then \( XY = YZ \), so \( 2x - 4 = x - 12 \), so \( x = -8 \). Then \( XY = 2(-8) - 4 = -20 \), \( YZ = -8 - 12 = -20 \). Then \( XZ = XY + YZ = -20 + (-20) = -40 \). But length can't be negative. Wait, maybe the problem was \( XY = 2x + 4 \) and \( YZ = x - 12 \)? Let's assume that maybe it's a typo and it's \( 2x + 4 \). Let's try that. Then \( 2x + 4 = x - 12 \), \( x = -16 \). Then \( XY = 2(-16) + 4 = -28 \), \( YZ = -16 - 12 = -28 \), still negative. Wait, maybe the problem is \( XY = 2x - 4 \) and \( YZ = x + 12 \)? Let's check the original problem again. The user wrote: "XY = 2x - 4; YZ = x - 12". Hmm. Maybe the problem is correct, and we just proceed with the math. So \( XZ = XY + YZ \), and since \( XY = YZ \), \( XZ = 2 \times XY \) or \( 2 \times YZ \).

So with \( x = -8 \), \( XY = -20 \), \( YZ = -20 \), so \( XZ = -20 + (-20) = -40 \). But length is usually positive, so maybe the problem has a sign error, but mathematically, following the steps:

Wait, maybe I misread the problem. Let's re-express:

Points \( X, Y, Z \) are collinear, \( Y \) bisects \( XZ \), so \( XY = YZ \), and \( XZ = XY + YZ = 2XY = 2YZ \).

Given \( XY = 2x - 4 \), \( YZ = x - 12 \).

Set \( 2x - 4 = x - 12 \)

Subtract \( x \) from both sides: \( x - 4 = -12 \)

Add 4 to both sides: \( x = -8 \)

Then \( XY = 2(-8) - 4 = -16 - 4 = -20 \)

\( YZ = -8 - 12 = -20 \)

Then \( XZ = XY + YZ = -20 + (-20) = -40 \)

But since length can't be negative, maybe the problem intended \( XY = 2x + 4 \) and \( YZ = x - 12 \), but as per the given problem, we proceed.

Answer:

\( \boxed{-40} \) (Note: The negative sign might indicate a direction or a mistake in the problem setup, but mathematically, this is the result.)