Question

are these scalene triangles similar?
p t s
14 49
r
28
8
q
yes
no

Explanation:

Step1: Identify corresponding sides

First, we need to find the ratios of the corresponding sides. Let's consider triangles \( \triangle PTR \) and \( \triangle PQS \) (assuming the lines are parallel, so by the Basic Proportionality Theorem or AA similarity, but here we check side ratios). The sides: \( PT = 14 \), \( PS = PT + TS = 14 + 49 = 63 \); \( PR = 28 \), \( PQ = PR + RQ = 28 + 8 = 36 \)? Wait, no, wait. Wait, actually, the triangles are \( \triangle PTR \) and \( \triangle PQS \)? Wait, no, looking at the diagram, \( T \) is on \( PS \), \( R \) is on \( PQ \)? Wait, no, the labels: \( P \) to \( T \) is 14, \( T \) to \( S \) is 49, so \( PS = 14 + 49 = 63 \). \( R \) to \( Q \) is 8, \( S \) to \( R \) is 28, so \( SQ = 28 + 8 = 36 \). Wait, no, maybe the triangles are \( \triangle PT R \) and \( \triangle P S Q \)? Wait, no, let's check the ratios of the sides. Let's take the sides adjacent to the common angle at \( P \). So \( PT / PS = 14 / (14 + 49) = 14 / 63 = 2 / 9 \). And \( PR / PQ = 28 / (28 + 8) = 28 / 36 = 7 / 9 \). Wait, that's not equal. Wait, maybe I got the sides wrong. Wait, maybe the triangles are \( \triangle T S R \) and \( \triangle P Q R \)? No, wait, the problem is about scalene triangles, so we need to check if the ratios of corresponding sides are equal. Wait, another approach: the lines \( TR \) and \( SQ \) are parallel? Wait, the diagram shows \( TR \) parallel to \( SQ \)? If so, then by AA similarity (since angle at \( P \) is common, and angle \( PTR \) equal to angle \( PSQ \) because of parallel lines), but we can also check the side ratios. Wait, let's list the sides:

For triangle \( PTR \): sides \( PT = 14 \), \( PR = 28 \), and \( TR \) (unknown). For triangle \( PSQ \): sides \( PS = 14 + 49 = 63 \), \( PQ = 28 + 8 = 36 \), and \( SQ \) (unknown). Wait, no, that can't be. Wait, maybe the triangles are \( \triangle TSR \) and \( \triangle PQR \)? No, I think I made a mistake. Wait, let's check the ratios of the segments. \( PT / TS = 14 / 49 = 2 / 7 \), and \( PR / RQ = 28 / 8 = 7 / 2 \). These are not equal. Wait, but similarity requires corresponding sides to be in proportion. Wait, maybe the triangles are \( \triangle PTR \) and \( \triangle SQR \)? No, that doesn't make sense. Wait, no, the correct way: to check if two triangles are similar, the ratios of corresponding sides must be equal. Let's take the sides:

\( PT = 14 \), \( TS = 49 \), so \( PT / TS = 14/49 = 2/7 \)

\( PR = 28 \), \( RQ = 8 \), so \( PR / RQ = 28/8 = 7/2 \)

These ratios are not equal, and also, if we consider the other sides, \( PT / PQ \) (wait, \( PQ = PR + RQ = 28 + 8 = 36 \), \( PT = 14 \), so 14/36 = 7/18; \( PS = PT + TS = 14 + 49 = 63 \), \( PR = 28 \), so 28/63 = 4/9. Not equal. Wait, but maybe I got the triangles wrong. Wait, the problem says "these scalene triangles" – probably \( \triangle PTR \) and \( \triangle PSQ \), with \( T \) on \( PS \) and \( R \) on \( PQ \). So \( PT = 14 \), \( PS = 14 + 49 = 63 \); \( PR = 28 \), \( PQ = 28 + 8 = 36 \). Then \( PT / PS = 14/63 = 2/9 \), \( PR / PQ = 28/36 = 7/9 \). Not equal. Wait, that can't be. Wait, maybe the sides are \( PT = 14 \), \( TS = 49 \), \( SR = 28 \), \( RQ = 8 \). So the triangles are \( \triangle PTR \) and \( \triangle SQR \)? No, angle at \( T \) and \( S \)? Wait, no, maybe the triangles are \( \triangle PTS \) and \( \triangle PRQ \)? No, I'm confused. Wait, let's calculate the ratios of the corresponding sides. Let's see: \( PT = 14 \), \( PR = 28 \), so \( PT/PR = 14/28 = 1/2 \). \( TS = 49 \), \( RQ = 8 \), so \( TS/RQ = 49/8 \approx 6.125 \). Not equal. Wait, another pair: \( PT = 14 \), \( RQ = 8 \); \( TS = 49 \), \( PR = 28 \). 14/8 = 7/4, 49/28 = 7/4. Oh! Wait, that's equal. Wait, maybe the corresponding sides are \( PT \) and \( RQ \), \( TS \) and \( PR \), and \( TR \) and \( SQ \)? Wait, let's check:

\( PT = 14 \), \( RQ = 8 \): 14/8 = 7/4

\( TS = 49 \), \( PR = 28 \): 49/28 = 7/4

Ah! So the ratios are equal. So by SSS similarity (if the third sides are also in ratio 7/4), or by SAS similarity (if the included angle is equal). Wait, the included angle between \( PT \) and \( TS \) is angle at \( T \), and between \( RQ \) and \( PR \) is angle at \( R \)? No, wait, maybe the triangles are \( \triangle PTS \) and \( \triangle RQP \)? Wait, no, the angle between \( PT \) and \( TS \) is angle \( T \), and between \( PR \) and \( RQ \) is angle \( R \). But if \( TR \) is parallel to \( SQ \), then angle \( T \) equals angle \( S \), and angle \( R \) equals angle \( Q \), but maybe the correct corresponding sides are \( PT/PR = 14/28 = 1/2 \), \( TS/RQ = 49/8 \), which is not equal. Wait, no, I think I messed up the correspondence. Let's do it properly:

To check similarity, the order of the letters matters. Let's assume triangle \( PTR \) and triangle \( PSQ \). Then:

\( PT / PS = 14 / (14 + 49) = 14 / 63 = 2/9 \)

\( PR / PQ = 28 / (28 + 8) = 28 / 36 = 7/9 \)

Not equal. So that's not similar.

Wait, but earlier, \( PT / RQ = 14 / 8 = 7/4 \), \( TS / PR = 49 / 28 = 7/4 \). So if the triangles are \( \triangle PTR \) and \( \triangle RQS \)? Wait, no, the labels. Maybe the triangles are \( \triangle TSR \) and \( \triangle PQR \)? No, this is confusing. Wait, the key is: for two triangles to be similar, the ratios of corresponding sides must be equal. Let's list all possible side ratios:

First triangle sides (assuming \( \triangle PTR \)): \( PT = 14 \), \( PR = 28 \), \( TR \) (let's say \( x \))

Second triangle sides ( \( \triangle TSQ \)): \( TS = 49 \), \( SQ = 28 + 8 = 36 \), \( TQ \) (no, \( SQ = 36 \)? Wait, \( SR = 28 \), \( RQ = 8 \), so \( SQ = 28 + 8 = 36 \). \( PT = 14 \), \( TS = 49 \), so \( PT/TS = 14/49 = 2/7 \). \( PR/SQ = 28/36 = 7/9 \). Not equal.

Wait, maybe the triangles are \( \triangle PTR \) and \( \triangle SQR \). Then \( PT = 14 \), \( SQ = 36 \); \( PR = 28 \), \( SR = 28 \); \( TR \) and \( QR = 8 \). No, \( SR = 28 \), \( PR = 28 \), so that's isoceles, but the problem says scalene. So that can't be.

Wait, I think I made a mistake in identifying the triangles. Let's look again: the diagram has \( P \) connected to \( T \) (14), \( T \) to \( S \) (49), \( S \) to \( R \) (28), \( R \) to \( Q \) (8), and \( P \) to \( R \), \( T \) to \( R \). So the two triangles are \( \triangle PTR \) (with sides \( PT = 14 \), \( PR = 28 \), \( TR \)) and \( \triangle PSQ \) (with sides \( PS = 14 + 49 = 63 \), \( PQ = 28 + 8 = 36 \), \( SQ \)). Now, check the ratios:

\( PT / PS = 14 / 63 = 2/9 \)

\( PR / PQ = 28 / 36 = 7/9 \)

These are not equal, so the sides are not in proportion. Therefore, the triangles are not similar. Wait, but earlier when I took \( PT / RQ = 14/8 = 7/4 \) and \( TS / PR = 49/28 = 7/4 \), that's equal, but those are not corresponding sides of the same triangles. So the correct corresponding sides should be from the same triangles. So since the ratios of the sides of the two triangles ( \( \triangle PTR \) and \( \triangle PSQ \)) are not equal, the triangles are not similar.

Step2: Conclusion

Since the ratios of the corresponding sides are not equal ( \( 14/63 = 2/9 \) and \( 28/36 = 7/9 \), which are not equal), the scalene triangles are not similar.

Answer:

no