Question

select the equation to solve for x, using the segment addition postulate. diagram: segment sv with points t, u (s---t (13)---u (6)---v (2x-18)), total length sv = 4x-29. options: 13 + 6 + (4x - 29) = (2x - 18); 13 + 6 + (2x - 18) = (4x - 29); 13 + 6 = (2x - 18) + (4x - 29)

Explanation:

The Segment Addition Postulate states that if three points \( S \), \( T \), \( U \), \( V \) are collinear with \( T \) and \( U \) between \( S \) and \( V \), then \( SV = ST + TU + UV \). From the diagram, \( ST = 13 \), \( TU = 6 \), \( UV = 2x - 18 \), and \( SV = 4x - 29 \). So by the Segment Addition Postulate, \( 13 + 6 + (2x - 18) = (4x - 29) \)? Wait, no, wait. Wait, actually, the total length \( SV \) should be equal to the sum of \( ST \), \( TU \), and \( UV \). Wait, let's re - examine:

Wait, the left - hand side of the equation should be the sum of the smaller segments, and the right - hand side should be the larger segment. Wait, \( ST = 13 \), \( TU = 6 \), \( UV = 2x - 18 \), and the entire segment \( SV=4x - 29 \). So according to the Segment Addition Postulate, \( ST + TU+UV = SV \). So \( 13 + 6+(2x - 18)=4x - 29 \)? Wait, no, the options are:

Option 1: \( 13 + 6+(4x - 29)=(2x - 18) \)

Option 2: \( 13 + 6+(2x - 18)=(4x - 29) \)

Option 3: \( 13 + 6=(2x - 18)+(4x - 29) \)

Wait, the Segment Addition Postulate is \( ST + TU+UV = SV \). Here, \( ST = 13 \), \( TU = 6 \), \( UV = 2x - 18 \), \( SV = 4x - 29 \). So \( 13+6+(2x - 18)=4x - 29 \), which is the second option.

Let's verify:

The Segment Addition Postulate: If a point \( T \) is between \( S \) and \( U \), and a point \( U \) is between \( T \) and \( V \), then \( ST + TU+UV=SV \).

Given \( ST = 13 \), \( TU = 6 \), \( UV = 2x - 18 \), \( SV = 4x - 29 \).

So substituting into the postulate: \( 13 + 6+(2x - 18)=4x - 29 \), which matches the second option (the blue - colored option: \( 13 + 6+(2x - 18)=(4x - 29) \))

Answer:

The correct equation is \( 13 + 6+(2x - 18)=(4x - 29) \) (the equation in the blue box)