Question

find ux.
ux =
submit

Explanation:

Step1: Identify Midline Theorem

Since \( VW \) and \( UT \) have equal markings, \( W \) and \( T \) are midpoints? Wait, no, the top segment is 80, \( WX = 48 \), so \( VW = 80 - 48 = 32 \)? Wait, actually, the line \( WT \) is parallel to \( VU \) (by the midline or basic proportionality theorem, Thales' theorem). So the triangles are similar. Let \( UX = x \), then \( UT = 30 \), \( UX = x \), so \( XT = x - 30 \)? Wait, no, the sides: \( VX \) is split by \( W \) into \( VW \) and \( WX \), and \( UX \) is split by \( T \) into \( UT \) and \( TX \). By the Basic Proportionality Theorem (Thales' theorem), if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So \( \frac{VW}{WX} = \frac{UT}{TX} \)? Wait, no, maybe \( VW + WX = VX = 80 \)? Wait, the top segment is 80, so \( VX = 80 \), and \( WX = 48 \), so \( VW = 80 - 48 = 32 \). Then, since \( WT \) is parallel to \( VU \), triangle \( WT X \) is similar to triangle \( VU X \)? Wait, no, \( WT \) is parallel to \( VU \), so angle \( V = \) angle \( W \), angle \( U = \) angle \( T \), so triangle \( VU X \) ~ triangle \( WT X \). So the ratio of sides: \( \frac{VW}{WX} = \frac{UT}{TX} \)? Wait, \( VW = 32 \), \( WX = 48 \), \( UT = 30 \), \( TX = UX - UT = UX - 30 \). So \( \frac{32}{48} = \frac{30}{UX - 30} \). Cross-multiplying: \( 32(UX - 30) = 48 \times 30 \). \( 32UX - 960 = 1440 \). \( 32UX = 2400 \). \( UX = \frac{2400}{32} = 75 \). Wait, let's check again. Alternatively, the ratio of \( VW \) to \( VX \) is \( 32/80 = 2/5 \), and the ratio of \( UT \) to \( UX \) should be the same? Wait, no, maybe the other way. Wait, the top length is 80, \( WX = 48 \), so \( VW = 80 - 48 = 32 \). The segment \( WT \) is parallel to \( VU \), so by Thales' theorem, \( \frac{VW}{WX} = \frac{UT}{TX} \). Wait, \( VW = 32 \), \( WX = 48 \), \( UT = 30 \), \( TX = UX - 30 \). So \( 32/48 = 30/(UX - 30) \). Simplify \( 32/48 = 2/3 \). So \( 2/3 = 30/(UX - 30) \). Cross-multiplying: \( 2(UX - 30) = 90 \). \( 2UX - 60 = 90 \). \( 2UX = 150 \). \( UX = 75 \). Yes, that works.

Step2: Apply Proportionality Theorem

Let’s define the segments: \( VX = 80 \) (the top segment), \( WX = 48 \), so \( VW = 80 - 48 = 32 \). Since \( WT \parallel VU \), by the Basic Proportionality Theorem (Thales’ theorem), \( \frac{VW}{WX} = \frac{UT}{TX} \). Let \( UX = x \), then \( TX = x - 30 \) (since \( UT = 30 \)). Substituting the known values: \( \frac{32}{48} = \frac{30}{x - 30} \). Simplify \( \frac{32}{48} = \frac{2}{3} \), so \( \frac{2}{3} = \frac{30}{x - 30} \). Cross - multiply: \( 2(x - 30) = 3 \times 30 \). \( 2x - 60 = 90 \). Add 60 to both sides: \( 2x = 150 \). Divide by 2: \( x = 75 \).

Answer:

\( 75 \)