Question

find ru.
u
v
17
r
16
t
8
s
ru =
submit

Explanation:

Step1: Identify Midline Theorem

The segments with arrows are midlines, so \( T \) and \( V \) are midpoints. Thus, \( UT = TS = 16 \) and \( UV = VR = 17 \)? Wait, no, wait. Wait, \( UT = 16 \), \( TS = 8 \)? Wait, no, \( U T = 16 \), \( T S = 8 \), so \( U S = UT + TS = 16 + 8 = 24 \)? Wait, no, the midline: in a triangle, the midline is parallel to the base and half its length. Wait, the midline connects midpoints of two sides. So \( TV \) is midline? Wait, the arrows on \( TV \) and \( SR \)? Wait, no, the two red arrows are on \( TV \) and \( SR \)? Wait, the diagram: \( U \), \( T \), \( S \) are on one side, \( U \), \( V \), \( R \) on the other. Wait, \( T \) is midpoint? Wait, \( UT = 16 \), \( TS = 8 \)? No, that can't be. Wait, maybe \( U T = 16 \), \( T S = 8 \), so \( U S = 16 + 8 = 24 \), and \( V \) is midpoint of \( UR \), \( T \) midpoint of \( US \)? Wait, no, the midline theorem: if \( T \) and \( V \) are midpoints, then \( TV \parallel SR \) and \( TV = \frac{1}{2} SR \), but here we need \( RU \). Wait, \( RU = RV + VU \). Wait, \( RV = 17 \), so if \( V \) is midpoint, then \( VU = RV = 17 \), so \( RU = 17 + 17 = 34 \)? Wait, no, wait the other side: \( UT = 16 \), \( TS = 8 \), so \( U S = 24 \), but maybe the ratio. Wait, \( UT / US = 16 / (16 + 8) = 16 / 24 = 2 / 3 \)? No, that's not. Wait, maybe \( T \) is midpoint? Wait, \( UT = 16 \), \( TS = 8 \), so \( UT = 2 \times TS \), so \( T \) is not midpoint. Wait, maybe the triangle is \( \triangle USR \), with midline \( TV \). So \( TV \) is midline, so \( TV \parallel SR \) and \( TV = \frac{1}{2} SR \), but we need \( RU \). Wait, \( RV = 17 \), so if \( V \) is midpoint, then \( RU = 2 \times RV \)? No, \( RU = RV + VU \), so if \( V \) is midpoint, \( VU = RV = 17 \), so \( RU = 34 \). Wait, but the other side: \( UT = 16 \), \( TS = 8 \), so \( U S = 24 \), but maybe the ratio of sides. Wait, \( UT / US = 16 / 24 = 2 / 3 \), so the triangles are similar with ratio 2:3? No, that's not. Wait, maybe I made a mistake. Wait, the problem is to find \( RU \). Let's re-examine: \( U \) to \( T \) is 16, \( T \) to \( S \) is 8, so \( U S = 24 \). \( V \) is on \( UR \), \( T \) on \( US \), and \( TV \) is midline. So by midline theorem, \( TV \parallel SR \) and \( TV = \frac{1}{2} SR \), but we need \( RU \). Wait, \( RV = 17 \), so if \( V \) is midpoint, then \( RU = 2 \times RV = 34 \). Alternatively, \( UT = 16 \), \( TS = 8 \), so \( UT = 2 \times TS \), so \( T \) divides \( US \) in ratio 2:1. Then by the basic proportionality theorem (Thales' theorem), \( \frac{UT}{TS} = \frac{UV}{VR} \). So \( \frac{16}{8} = \frac{UV}{17} \), so \( 2 = \frac{UV}{17} \), so \( UV = 34 \)? No, wait, \( \frac{UT}{TS} = \frac{UV}{VR} \), so \( \frac{16}{8} = \frac{UV}{17} \), so \( UV = 34 \), then \( RU = UV + VR = 34 + 17 = 51 \)? Wait, that can't be. Wait, no, 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 \( TV \parallel SR \), so \( \frac{UT}{TS} = \frac{UV}{VR} \). \( UT = 16 \), \( TS = 8 \), so \( \frac{16}{8} = 2 \), so \( \frac{UV}{VR} = 2 \), so \( UV = 2 \times VR = 2 \times 17 = 34 \), then \( RU = UV + VR = 34 + 17 = 51 \)? Wait, but that contradicts the midline. Wait, maybe the diagram is \( \triangle USR \), with \( U \), \( T \), \( S \) on one side, \( U \), \( V \), \( R \) on the other. So \( UT = 16 \), \( TS = 8 \), so \( US = 24 \). \( TV \) is parallel to \( SR \), so \( \triangle UTV \sim \triangle USR \) by AA similarity. The ratio of sides \( UT / US = 16 / 24 = 2 / 3 \), so the ratio of \( UV / UR = 2 / 3 \). But \( UV = UR - VR = UR - 17 \), so \( (UR - 17) / UR = 2 / 3 \), solving: \( 3(UR - 17) = 2 UR \), \( 3 UR - 51 = 2 UR \), \( UR = 51 \). Wait, that makes sense. So:

Step1: Apply Similar Triangles

Since \( TV \parallel SR \), \( \triangle UTV \sim \triangle USR \) (AA similarity, as \( \angle U \) is common and \( \angle UTV = \angle USR \) (corresponding angles)).

Step2: Set Up Proportion

The ratio of sides \( \frac{UT}{US} = \frac{UV}{UR} \). \( UT = 16 \), \( US = UT + TS = 16 + 8 = 24 \), \( UV = UR - VR = UR - 17 \). So:
\[ \frac{16}{24} = \frac{UR - 17}{UR} \]

Step3: Solve for \( UR \)

Simplify \( \frac{16}{24} = \frac{2}{3} \), so:
\[ \frac{2}{3} = \frac{UR - 17}{UR} \]
Cross-multiply: \( 2 \times UR = 3 \times (UR - 17) \)
\[ 2UR = 3UR - 51 \]
Subtract \( 2UR \) from both sides: \( 0 = UR - 51 \)
So \( UR = 51 \).

Answer:

\( 51 \)