Question

find ru.
r s t
44 22
v
10
u
ru =
submit

Explanation:

Step1: Identify Midline Theorem

The segment \( SV \) is a midline (since the arrows indicate it's parallel and bisecting the sides), so by the Midline Theorem, \( SV \parallel RU \) and \( ST = RS \) implies \( VU = RV \). Wait, actually, the Midline Theorem (or Basic Proportionality Theorem/Thales' theorem) states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Here, \( SV \) is parallel to \( RU \) (implied by the midline marks), so \( \frac{RS}{RT} = \frac{RV}{RU} \). First, find \( RT = RS + ST = 44 + 22 = 66 \). Also, \( VU = 10 \), and since \( SV \) is a midline, \( RV = VU \)? Wait, no, let's check the ratios. \( RS = 44 \), \( ST = 22 \), so \( RS = 2 \times ST \), meaning the ratio of \( RS \) to \( RT \) is \( \frac{44}{44 + 22} = \frac{44}{66} = \frac{2}{3} \)? Wait, no, actually, if \( SV \) is parallel to \( RU \), then triangle \( RSV \) is similar to triangle \( RTU \) by AA similarity (since \( \angle R \) is common and \( \angle RSV = \angle RTU \) because of parallel lines). So the ratio of similarity is \( \frac{RS}{RT} = \frac{44}{44 + 22} = \frac{44}{66} = \frac{2}{3} \)? Wait, no, \( RS = 44 \), \( RT = 44 + 22 = 66 \), so \( RS/RT = 44/66 = 2/3 \). But also, \( RV/RU = RS/RT \). Wait, \( VU = 10 \), so \( RU = RV + VU \). Let \( RV = x \), then \( RU = x + 10 \). From similarity, \( x/(x + 10) = 44/66 = 2/3 \). Cross-multiplying: \( 3x = 2(x + 10) \) → \( 3x = 2x + 20 \) → \( x = 20 \). Then \( RU = x + 10 = 20 + 10 = 30 \)? Wait, no, wait, maybe I mixed up the sides. Wait, actually, the segment \( SV \) is connecting \( S \) on \( RT \) and \( V \) on \( RU \), with \( ST = 22 \), \( RS = 44 \), so \( RS = 2 \times ST \), meaning \( S \) divides \( RT \) into a ratio of \( 2:1 \) (RS:ST = 44:22 = 2:1). Therefore, by the Basic Proportionality Theorem, \( V \) should divide \( RU \) into the same ratio? Wait, no, the theorem says that if a line is parallel to one side, it divides the other two sides proportionally. So \( \frac{RS}{ST} = \frac{RV}{VU} \). Ah, that's the correct ratio! Because \( SV \parallel TU \)? Wait, no, the triangle is \( R - U - T \), with \( S \) on \( RT \) and \( V \) on \( RU \), and \( SV \) parallel to \( TU \)? Wait, maybe the diagram is a triangle \( RTU \), with \( S \) on \( RT \) (RS = 44, ST = 22) and \( V \) on \( RU \) (VU = 10), and \( SV \) parallel to \( TU \). Then by the Basic Proportionality Theorem (Thales' theorem), \( \frac{RS}{ST} = \frac{RV}{VU} \). So \( \frac{44}{22} = \frac{RV}{10} \) → \( 2 = \frac{RV}{10} \) → \( RV = 20 \). Then \( RU = RV + VU = 20 + 10 = 30 \). Wait, that makes sense. Because \( RS/ST = 44/22 = 2 \), so \( RV/VU = 2 \), so \( RV = 2 \times VU = 2 \times 10 = 20 \), so \( RU = 20 + 10 = 30 \).

Step2: Apply Thales' Theorem

Using the Basic Proportionality Theorem (Thales' theorem) in triangle \( RTU \), since \( SV \parallel TU \), we have \( \frac{RS}{ST} = \frac{RV}{VU} \). Given \( RS = 44 \), \( ST = 22 \), \( VU = 10 \). Substitute the values: \( \frac{44}{22} = \frac{RV}{10} \). Simplify \( \frac{44}{22} = 2 \), so \( 2 = \frac{RV}{10} \). Solve for \( RV \): \( RV = 2 \times 10 = 20 \). Then \( RU = RV + VU = 20 + 10 = 30 \).

Answer:

\( 30 \)