Question

find rs.
rs =
work it out
not feeling ready yet? these can help.

Explanation:

Step1: Identify the Midline Theorem

Since \( QS \) is a midline (as the marks indicate midpoints), by the Midline Theorem, \( QS \parallel UT \) and \( QS=\frac{1}{2}UT \), and also the triangles are similar. So the ratio of corresponding sides is equal. Let \( RS = x \), then \( RT=RS + ST=x + 38 \), and the ratio of \( RQ \) to \( RU \) should equal the ratio of \( RS \) to \( RT \). \( RQ = 45 - 30=15 \)? Wait, no, looking at the diagram, \( RU = 45 \), \( QU = 30 \), so \( RQ=45 - 30 = 15 \)? Wait, no, maybe the segments: \( RU \) is the whole side, \( QU = 30 \), \( RQ=45 - 30=15 \)? Wait, no, actually, the midline is \( QS \), so \( Q \) is the midpoint of \( RU \) and \( S \) is the midpoint of \( RT \)? Wait, no, the marks on \( QU \) and the other side ( \( UT \)?) Wait, the diagram has \( QU = 30 \), \( UT \) side? Wait, maybe the correct approach is using the Basic Proportionality Theorem (Thales' theorem) or similar triangles. Let's assume that \( \triangle RQS \sim \triangle RUT \) by AA similarity (since \( QS \parallel UT \), corresponding angles are equal). So the ratio of sides \( \frac{RQ}{RU}=\frac{RS}{RT} \). From the diagram, \( RU = 45 \), \( RQ = 45 - 30 = 15 \)? Wait, no, maybe \( RU = 45 \), \( QU = 30 \), so \( RQ=45 - 30 = 15 \), and \( RT=RS + ST=RS + 38 \). Wait, no, maybe \( ST = 38 \), and \( S \) is the midpoint? Wait, the mark on the side \( UT \) (the left side) and on \( RT \) (the right side) indicates midpoints. So \( Q \) is the midpoint of \( RU \), so \( RQ = QU = 30 \)? Wait, that can't be, because \( RU = 45 \). Wait, maybe the diagram is labeled as: \( RU = 45 \), \( QU = 30 \), so \( RQ=45 - 30 = 15 \), and \( ST = 38 \), \( S \) is a point on \( RT \), \( Q \) is a point on \( RU \), and \( QS \parallel UT \). So by Thales' theorem, \( \frac{RQ}{RU}=\frac{RS}{RT} \). So \( RQ = 15 \), \( RU = 45 \), \( RS = x \), \( RT=x + 38 \). So \( \frac{15}{45}=\frac{x}{x + 38} \).

Step2: Solve the Proportion

Simplify \( \frac{15}{45}=\frac{1}{3} \), so \( \frac{1}{3}=\frac{x}{x + 38} \). Cross - multiply: \( x + 38 = 3x \). Subtract \( x \) from both sides: \( 38 = 2x \). Then \( x=\frac{38}{2}=19 \). Wait, that makes sense. Because if \( QS \) is the midline, but maybe the ratio is \( 1:3 \)? Wait, no, let's re - check. If \( RQ = 15 \), \( RU = 45 \), so the ratio is \( 15:45 = 1:3 \). Then \( RS:RT=1:3 \), and \( RT=RS + 38 \), so \( RS:(RS + 38)=1:3 \), \( 3RS=RS + 38 \), \( 2RS = 38 \), \( RS = 19 \).

Answer:

\( 19 \)