Question

qu || rt. find qs. qs =
submit

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{QU} \parallel \overline{RT} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle QUS \sim \triangle RTS \). So the ratios of corresponding sides are equal: \( \frac{QS}{RS}=\frac{US}{TS} \). Wait, actually, let's define the segments. Let \( QS = x \), then \( RS = x - 30 \)? Wait no, looking at the diagram: \( QR = 30 \), \( RS \) is unknown? Wait no, the horizontal segment: \( Q \) to \( R \) is 30, \( R \) to \( S \) is let's say \( y \), so \( QS = 30 + y \). The vertical segment: \( T \) to \( S \) is 10, \( U \) to \( S \) is \( 10 + 30 = 40 \)? Wait no, the right side: the total length from \( U \) to \( S \) (along the vertical) is \( 30 + 10 = 40 \)? Wait the diagram has a vertical line with length 30 (from the top mark to \( T \)) and 10 from \( T \) to \( S \), so \( US = 30 + 10 = 40 \)? Wait no, maybe the vertical segments: \( TS = 10 \), \( US = 30 \)? Wait no, the diagram shows a vertical line with two segments: 30 (from the top tick to \( T \)) and 10 (from \( T \) to \( S \)), so \( US \) is the length from \( U \) to \( S \), which is \( 30 + 10 = 40 \)? Wait no, maybe \( QU \parallel RT \), so \( \triangle QUS \) and \( \triangle RTS \) are similar. So \( \frac{QS}{RS}=\frac{US}{TS} \). Wait, \( US \) is the length from \( U \) to \( S \), \( TS \) is from \( T \) to \( S \). Wait the vertical side: \( TS = 10 \), \( US = 10 + 30 = 40 \)? Wait no, the diagram's right side: the top segment (from the tick to \( U \)?) No, the tick marks: the vertical line has two segments, 30 (from the upper tick to \( T \)) and 10 (from \( T \) to \( S \)), so \( UT = 30 \), \( TS = 10 \), so \( US = UT + TS = 30 + 10 = 40 \). The horizontal line: \( QR = 30 \), \( RS \) is the segment from \( R \) to \( S \), and \( QS = QR + RS = 30 + RS \). Now, since \( QU \parallel RT \), \( \triangle QUS \sim \triangle RTS \) (by AA similarity, as \( \angle QSU \) is common and \( \angle QUS = \angle RTS \) because of parallel lines). So the ratio of corresponding sides: \( \frac{QS}{RS}=\frac{US}{TS} \). Let \( RS = y \), then \( QS = 30 + y \), \( US = 40 \), \( TS = 10 \). So \( \frac{30 + y}{y}=\frac{40}{10} \).

Step2: Solve the Proportion

Simplify \( \frac{40}{10}=4 \), so \( \frac{30 + y}{y}=4 \). Multiply both sides by \( y \): \( 30 + y = 4y \). Subtract \( y \) from both sides: \( 30 = 3y \). Divide by 3: \( y = 10 \). Then \( QS = 30 + y = 30 + 10 = 40 \)? Wait no, that can't be. Wait maybe I mixed up the segments. Wait the horizontal: \( Q \) to \( R \) is 30, \( R \) to \( S \) is \( x \), so \( QS = 30 + x \). The vertical: \( T \) to \( S \) is 10, \( U \) to \( T \) is 30, so \( U \) to \( S \) is \( 30 + 10 = 40 \). Wait the similar triangles: \( \triangle RT S \) and \( \triangle QU S \). So \( \frac{RT}{QU}=\frac{TS}{US}=\frac{RS}{QS} \). Wait maybe the correct proportion is \( \frac{QR}{QS}=\frac{TS}{US} \)? No, let's re-examine. Since \( QU \parallel RT \), \( \angle Q = \angle TRS \) (corresponding angles), and \( \angle S \) is common. So \( \triangle QUS \sim \triangle RTS \). Therefore, \( \frac{QS}{RS}=\frac{US}{TS} \). Wait \( US \) is \( UT + TS = 30 + 10 = 40 \), \( TS = 10 \), \( RS \) is the segment from \( R \) to \( S \), \( QS = QR + RS = 30 + RS \). So \( \frac{30 + RS}{RS}=\frac{40}{10}=4 \). Then \( 30 + RS = 4RS \), \( 30 = 3RS \), \( RS = 10 \), so \( QS = 30 + 10 = 40 \). Wait but that seems off. Wait maybe the vertical segment: the 30 is from \( U \) to the top tick, and \( T \) to \( S \) is 10, so \( US = 30 \) and \( TS = 10 \)? Wait the diagram: the right side has a segment with length 30 (from the top tick to \( T \)) and 10 (from \( T \) to \( S \)), so \( UT = 30 \), \( TS = 10 \), so \( US = UT + TS = 40 \)? No, maybe the 30 is from \( U \) to \( S \), and 10 from \( T \) to \( S \), so \( UT = 30 - 10 = 20 \)? Wait I think I misread the diagram. Let's look again: the horizontal line: \( Q \)---30---\( R \)---?---\( S \). The vertical line: \( U \)---?---\( T \)---10---\( S \), with the total vertical length (from \( U \) to the top tick) being 30? No, the top tick is next to \( U \)'s vertical line, with length 30. So the vertical segment from \( U \) to \( S \) is \( 30 + 10 \)? No, the diagram shows: on the right, a vertical line with two parts: 30 (from the top tick to \( T \)) and 10 (from \( T \) to \( S \)), so \( U \) is at the top of the blue triangle, \( T \) is a point on \( US \), \( S \) is the bottom. So \( US = UT + TS = 30 + 10 = 40 \). The horizontal: \( Q \) to \( R \) is 30, \( R \) to \( S \) is \( x \), so \( QS = 30 + x \). The triangles: \( \triangle QUS \) and \( \triangle RTS \) are similar. So \( \frac{QS}{RS}=\frac{US}{TS} \) → \( \frac{30 + x}{x}=\frac{40}{10}=4 \). Then \( 30 + x = 4x \) → \( 3x = 30 \) → \( x = 10 \). So \( QS = 30 + 10 = 40 \). Wait but that would mean \( RS = 10 \), so \( QS = 40 \). Alternatively, maybe the proportion is \( \frac{QR}{RS}=\frac{UT}{TS} \). \( QR = 30 \), \( RS = x \), \( UT = 30 \), \( TS = 10 \). So \( \frac{30}{x}=\frac{30}{10} \) → \( \frac{30}{x}=3 \) → \( x = 10 \). Then \( QS = QR + RS = 30 + 10 = 40 \). Yes, that makes sense. So the length of \( QS \) is 40.

Answer:

40