Question

find qu.
s t
45
70 r u 56
q
qu =
submit

Explanation:

Step1: Identify Similar Triangles

Since \( SR \parallel TU \) (indicated by the arrows) and \( QR \) and \( QU \) are parts of the sides, triangles \( QSR \) and \( QTU \) are similar by the Basic Proportionality Theorem (Thales' theorem) or AA similarity (common angle at \( Q \) and parallel lines imply equal corresponding angles).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{QS}{QT}=\frac{SR}{TU}=\frac{QR}{QU}\). We know \( QS = 70 \), \( SR = 45 \), and \( QT = 56 \)? Wait, no, wait: Wait, the sides with tick marks are equal? Wait, no, the left side: \( QS \) is 70 (from Q to S), and the segment from S to R is 45? Wait, no, maybe \( QS = 70 \), \( SR = 45 \), so \( QR = QS - SR = 70 - 45 = 25 \)? Wait, no, maybe I misread. Wait, the diagram: \( QS \) is a side with length 70? Wait, no, the left side: the two segments are 45 (from S to R) and 70 (from Q to R)? Wait, no, the tick marks: the left side (QS) and the right side (QT) have tick marks, meaning \( QS = QT \)? Wait, no, the left side has a tick mark, right side has a tick mark, so \( QS = QT \)? Wait, no, the left side: length from Q to S is 70? Wait, no, the numbers: 45 is the length of SR, 70 is the length of QR? Wait, maybe \( QS = QR + RS = 70 + 45 = 115 \)? No, that doesn't make sense. Wait, let's re-examine.

Wait, the correct approach: The triangles \( QSR \) and \( QTU \) are similar. So the ratio of \( QR \) to \( QU \) should be equal to the ratio of \( QS \) to \( QT \). Wait, no, maybe \( QS = 70 \), \( QT = 56 \), and \( SR = 45 \). Wait, no, let's look at the sides:

Wait, the side from Q to S is 70 (the left side with a tick mark), the side from Q to T is 56 (the right side with a tick mark)? No, tick marks usually mean equal length, but here the numbers are different. Wait, maybe the segments: \( QR = 70 - 45 = 25 \)? No, that's not right. Wait, maybe \( QS = 70 \), \( SR = 45 \), so \( QR = QS - SR = 70 - 45 = 25 \). Then \( QT = 56 \), and we need to find \( QU \). Since \( SR \parallel TU \), triangles \( QSR \sim QTU \), so \(\frac{QR}{QU}=\frac{QS}{QT}\). Wait, no, \( QS \) is 70, \( QT \) is 56? Wait, that can't be. Wait, maybe \( QS = 70 \), \( QT = 56 \), and \( SR = 45 \), \( TU = QU \)? No, let's do it properly.

Wait, the key is that \( RU \parallel ST \), so by the Basic Proportionality Theorem, \( \frac{QR}{QS} = \frac{QU}{QT} \). Wait, \( QR \) is the segment from Q to R, \( QS \) is from Q to S. So \( QR = 70 - 45 = 25 \)? No, wait, the length from Q to R is 70? No, the number 70 is next to the segment from Q to R, and 45 is from S to R. So \( QS = QR + RS = 70 + 45 = 115 \)? No, that's not matching. Wait, maybe the diagram is: \( QS = 70 \), \( SR = 45 \), so \( QR = QS - SR = 70 - 45 = 25 \). Then \( QT = 56 \), and we need to find \( QU \). Since \( SR \parallel TU \), triangles \( QSR \) and \( QTU \) are similar, so \( \frac{QR}{QU} = \frac{QS}{QT} \). Wait, no, \( QS \) is 70, \( QT \) is 56? That would give a ratio greater than 1, but \( QU \) should be less than \( QT \). Wait, maybe I have the ratio reversed. It should be \( \frac{QR}{QS} = \frac{QU}{QT} \). So \( QR = 70 - 45 = 25 \)? No, that's not. Wait, maybe \( QS = 70 \), \( QR = 45 \), and \( QT = 56 \). Then \( \frac{QR}{QS} = \frac{QU}{QT} \), so \( \frac{45}{70} = \frac{QU}{56} \). Then \( QU = \frac{45 \times 56}{70} \). Let's calculate that: \( 45 \times 56 = 2520 \), \( 2520 \div 70 = 36 \). Ah, that makes sense. So \( QR = 45 \), \( QS = 70 \), \( QT = 56 \). So the ratio of \( QR \) to \( QS \) is \( \frac{45}{70} \), and this should equal the ratio of \( QU \) to \( QT \), so \( \frac{QU}{56} = \frac{45}{70} \). Solving for \( QU \): \( QU = 56 \times \frac{45}{70} = 56 \times \frac{9}{14} = 4 \times 9 = 36 \).

Step3: Calculate QU

Using the proportion \(\frac{QR}{QS} = \frac{QU}{QT}\), substitute \( QR = 45 \), \( QS = 70 \), \( QT = 56 \):

\( QU = \frac{45 \times 56}{70} \)

Simplify \( \frac{56}{70} = \frac{4}{5} \)? Wait, no, \( 56 \div 70 = 0.8 \), and \( 45 \times 0.8 = 36 \). So \( QU = 36 \).

Answer:

36