Question

find qr.
t s
p r 22
30 19
q
qr =
submit

Explanation:

Step1: Identify Similar Triangles

Since \( PT \parallel PS \) (wait, actually \( PT \) and \( PR \) are parallel to \( TS \)? Wait, the lines \( TS \) and \( PR \) are parallel (since they have the same direction arrows), so triangles \( QPR \) and \( QTS \) are similar by the Basic Proportionality Theorem (Thales' theorem) or AA similarity (since corresponding angles are equal as \( PR \parallel TS \)).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{QP}{QT} = \frac{QR}{QS}\). We know \( QP = 19 \), \( QT = 30 \), and \( QS = QR + RS = QR + 22 \). Wait, no, actually \( QT \) is 30, \( QP \) is 19, \( QS \) is \( QR + 22 \)? Wait, no, looking at the diagram: \( QT \) is the side from \( Q \) to \( T \), length 30; \( QP \) is from \( Q \) to \( P \), length 19; \( QS \) is from \( Q \) to \( S \), with \( RS = 22 \), so \( QS = QR + 22 \). Since \( PR \parallel TS \), triangles \( QPR \sim QTS \), so \(\frac{QP}{QT} = \frac{QR}{QS}\). Let \( QR = x \), then \( QS = x + 22 \). So \(\frac{19}{30} = \frac{x}{x + 22}\).

Step3: Solve the Proportion

Cross-multiplying: \( 19(x + 22) = 30x \)
\( 19x + 418 = 30x \)
\( 418 = 30x - 19x \)
\( 418 = 11x \)
\( x = \frac{418}{11} = 38 \)? Wait, no, wait, maybe I mixed up the sides. Wait, maybe \( QT \) is 30, \( QP \) is 19, so the ratio of similarity is \( \frac{QP}{QT} = \frac{19}{30} \), but maybe \( QS \) is 22? Wait, no, the diagram: \( S \) to \( R \) is 22, so \( QS = QR + RS \), \( RS = 22 \). Wait, maybe I got the triangles wrong. Let's re-examine: \( T \) to \( S \) is a line, \( P \) to \( R \) is a line parallel to \( T \) to \( S \), so \( \triangle QPR \sim \triangle QTS \), so \( \frac{QP}{QT} = \frac{QR}{QS} \). Wait, \( QT \) is 30, \( QP \) is 19, \( QS \) is \( QR + 22 \)? No, maybe \( QS \) is 22? Wait, the length from \( Q \) to \( S \) is \( QR + RS \), and \( RS = 22 \). Wait, maybe the sides are \( QT = 30 \), \( QP = 19 \), \( QS = 22 + QR \)? Wait, no, let's check the numbers again. Wait, maybe the correct proportion is \( \frac{QP}{QT} = \frac{QR}{QS} \), where \( QT = 30 \), \( QP = 19 \), \( QS = 22 \)? No, that can't be. Wait, maybe I flipped the ratio. Since \( PR \parallel TS \), then \( \frac{QP}{QT} = \frac{QR}{QS} \), but \( QT = QP + PT \)? No, \( QT \) is a side, \( QP \) is a segment on \( QT \). Wait, \( QT \) is 30, \( QP \) is 19, so \( PT = QT - QP = 30 - 19 = 11 \). Then, since \( PR \parallel TS \), the ratio of \( QP \) to \( QT \) is \( 19/30 \), and the ratio of \( QR \) to \( QS \) should be the same. But \( QS = QR + RS \), \( RS = 22 \). Wait, no, maybe \( QS \) is 22, and \( QR \) is what we need. Wait, this is confusing. Wait, let's use the Basic Proportionality Theorem (Thales' theorem), which 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. So in \( \triangle QTS \), line \( PR \) is parallel to \( TS \), so \( \frac{QP}{PT} = \frac{QR}{RS} \). Wait, that makes more sense! Because \( PR \parallel TS \), so by Thales' theorem, \( \frac{QP}{PT} = \frac{QR}{RS} \). Let's check: \( QT = 30 \), \( QP = 19 \), so \( PT = QT - QP = 30 - 19 = 11 \). Then \( RS = 22 \), so \( \frac{QP}{PT} = \frac{19}{11} \), and \( \frac{QR}{RS} = \frac{QR}{22} \). Wait, no, Thales' theorem is \( \frac{QP}{QT} = \frac{QR}{QS} \), but maybe I had the segments wrong. Wait, let's start over.

Correct approach: In triangle \( QTS \), \( PR \parallel TS \), so by the Basic Proportionality Theorem (Thales' theorem), \( \frac{QP}{QT} = \frac{QR}{QS} \). Here, \( QP = 19 \), \( QT = 30 \), \( QS = QR + RS \), and \( RS = 22 \). So:

\(\frac{19}{30} = \frac{QR}{QR + 22}\)

Cross-multiplying:

\(19(QR + 22) = 30 \times QR\)

\(19QR + 418 = 30QR\)

\(418 = 30QR - 19QR\)

\(418 = 11QR\)

\(QR = \frac{418}{11} = 38\)? Wait, that can't be, because 11*38=418. Wait, but let's check the ratio. If \( QR = 38 \), then \( QS = 38 + 22 = 60 \). Then \( 19/30 = 38/60 \), which simplifies to \( 19/30 = 19/30 \), so that works. Oh! Wait, 38 + 22 = 60, and 19/30 = 38/60 (since 38 is 19*2, 60 is 30*2). So that's correct. So \( QR = 38 \)? Wait, no, wait, 19*2=38, 30*2=60, and 60 - 38=22, which is RS. So yes, that makes sense. So the proportion is correct.

Answer:

\( \boxed{38} \) Wait, no, wait, 418 divided by 11 is 38? Wait, 11*38=418, yes. So QR is 38? Wait, but let's check again. If QT is 30, QP is 19, then the ratio of similarity is 19/30. Then QS should be QR + 22, and QR/QS = 19/30. So QR/(QR + 22) = 19/30. Cross-multiplying: 30QR = 19QR + 418 → 11QR = 418 → QR = 418/11 = 38. Yes, that's correct. So the answer is 38.