Question

dy guide

1. is tu || qr? explain your reasoning

given:
st = 4, tq = 12, sr = 25, su = 5

Explanation:

Step1: Calculate \( \frac{ST}{SQ} \)

First, find \( SQ \) by adding \( ST \) and \( TQ \). So \( SQ = ST + TQ = 4 + 12 = 16 \). Then, \( \frac{ST}{SQ}=\frac{4}{16}=\frac{1}{4} \).

Step2: Calculate \( \frac{SU}{SR} \)

We know \( SR = 25 \) and \( SU = 5 \), so \( \frac{SU}{SR}=\frac{5}{25}=\frac{1}{5} \). Wait, no, wait. Wait, the 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. So we should check \( \frac{ST}{TQ} \) and \( \frac{SU}{UR} \)? Wait, no, let's re - examine the triangle. The triangle is \( \triangle SQR \), with point \( T \) on \( SQ \) and point \( U \) on \( SR \). So according to Thales' theorem, if \( TU\parallel QR \), then \( \frac{ST}{TQ}=\frac{SU}{UR} \). First, calculate \( UR = SR - SU=25 - 5 = 20 \). Then \( \frac{ST}{TQ}=\frac{4}{12}=\frac{1}{3} \), and \( \frac{SU}{UR}=\frac{5}{20}=\frac{1}{4} \). Wait, that's not equal. Wait, maybe I mixed up the segments. Wait, another way: \( \frac{ST}{SQ}=\frac{4}{4 + 12}=\frac{4}{16}=\frac{1}{4} \), and \( \frac{SU}{SR}=\frac{5}{25}=\frac{1}{5} \). Wait, no, that's not right. Wait, maybe the correct ratio is \( \frac{ST}{SQ}=\frac{SU}{SR} \)? Wait, no, let's recall the theorem again. The theorem is: In \( \triangle ABC \), if \( DE\parallel BC \), \( D \) on \( AB \), \( E \) on \( AC \), then \( \frac{AD}{DB}=\frac{AE}{EC} \). So in our case, \( \triangle SQR \), \( T \) on \( SQ \), \( U \) on \( SR \). So \( SQ = ST+TQ = 16 \), \( SR = SU + UR=25 \). So \( \frac{ST}{SQ}=\frac{4}{16}=\frac{1}{4} \), \( \frac{SU}{SR}=\frac{5}{25}=\frac{1}{5} \). Wait, these are not equal. Wait, maybe I made a mistake. Wait, no, let's check the ratio of \( ST/SQ \) and \( SU/SR \) again. Wait, \( ST = 4 \), \( SQ=4 + 12 = 16 \), so \( ST/SQ = 4/16=1/4 \). \( SU = 5 \), \( SR = 25 \), so \( SU/SR = 5/25 = 1/5 \). Since \( 1/4 eq1/5 \), does that mean \( TU \) is not parallel? Wait, no, maybe I got the segments wrong. Wait, maybe the ratio should be \( ST/TQ \) and \( SU/UR \). Let's calculate \( ST/TQ=\frac{4}{12}=\frac{1}{3} \), \( UR=SR - SU = 25 - 5 = 20 \), \( SU/UR=\frac{5}{20}=\frac{1}{4} \). These are not equal. Wait, but maybe I messed up the triangle structure. Wait, the given lengths: \( ST = 4 \), \( TQ = 12 \), so \( SQ=16 \); \( SU = 5 \), \( SR = 25 \), so \( UR = 20 \). Now, let's check \( \frac{ST}{SQ}=\frac{4}{16}=\frac{1}{4} \) and \( \frac{SU}{SR}=\frac{5}{25}=\frac{1}{5} \). Since these ratios are not equal, by the converse of Thales' theorem, \( TU \) is not parallel to \( QR \). Wait, but wait, maybe I had the ratio reversed. Let's check \( \frac{TQ}{ST}=\frac{12}{4} = 3 \) and \( \frac{UR}{SU}=\frac{20}{5}=4 \). Still not equal. So the conclusion is that \( TU \) is not parallel to \( QR \) because the ratios of the divided segments are not equal.

Wait, no, wait a second. I think I made a mistake in identifying the sides. Let's re - express the triangle. The triangle is \( S - Q - R \), with \( T \) on \( SQ \) (between \( S \) and \( Q \)) and \( U \) on \( SR \) (between \( S \) and \( R \)). So according to the converse of Thales' theorem, if \( \frac{ST}{SQ}=\frac{SU}{SR} \), then \( TU\parallel QR \). But \( \frac{ST}{SQ}=\frac{4}{4 + 12}=\frac{4}{16}=\frac{1}{4} \), \( \frac{SU}{SR}=\frac{5}{25}=\frac{1}{5} \). Since \( \frac{1}{4} eq\frac{1}{5} \), the lines are not parallel. Alternatively, if we consider \( \frac{ST}{TQ}=\frac{4}{12}=\frac{1}{3} \) and \( \frac{SU}{UR}=\frac{5}{20}=\frac{1}{4} \), and since \( \frac{1}{3} eq\frac{1}{4} \), by the converse of Thales' theorem, \( TU \) is not parallel to \( QR \).

Answer:

To determine if \( TU\parallel QR \), we use the converse of the Basic Proportionality Theorem (Thales' theorem).

1. First, find the lengths of the relevant segments:

• \( SQ=ST + TQ=4 + 12 = 16\)
• \( UR=SR - SU=25 - 5 = 20\)

1. Then, check the ratios:

• The ratio of the segments on \( SQ \): \( \frac{ST}{TQ}=\frac{4}{12}=\frac{1}{3}\)
• The ratio of the segments on \( SR \): \( \frac{SU}{UR}=\frac{5}{20}=\frac{1}{4}\)

Since \( \frac{ST}{TQ}
eq\frac{SU}{UR} \) (or \( \frac{ST}{SQ}
eq\frac{SU}{SR} \)), by the converse of the Basic Proportionality Theorem, \( TU \) is not parallel to \( QR \).

So the answer is \( TU \) is not parallel to \( QR \) (because the ratios of the divided sides are not equal, violating the converse of Thales' theorem).