Question

which figures show that \\(\overline{st} \parallel \overline{qr}\\)?

(first triangle: p at top, q and r at base. ps = 6.6, pq = 6.6 + 23.4 = 30; pt = 5.5, pr = 5.5 + 19.5 = 25. st is a segment between s on pq and t on pr.)

(second triangle: p at top, q and r at base. ps = 7.5, pq = 7.5 + 17.5 = 25; pt = 8.7, pr = 8.7 + 20.3 = 29. st is a segment between s on pq and t on pr.)

(third triangle: p at top, q and r at base. ps = 9, pq = 9 + 16 = 25; pt = 11.2, pr = 11.2 + 17.8 = 29. st is a segment between s on pq and t on pr.)

Explanation:

Step1: Recall the Basic Proportionality Theorem (Thales' theorem)

The 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, for \( \overline{ST} \parallel \overline{QR} \), we should have \( \frac{PS}{PQ} = \frac{PT}{PR} \) (or \( \frac{PS}{SQ} = \frac{PT}{TR} \)).

Step2: Analyze the first triangle

- \( PS = 6.6 \), \( PQ = PS + SQ = 6.6 + 23.4 = 30 \)
- \( PT = 5.5 \), \( PR = PT + TR = 5.5 + 19.5 = 25 \)
- Check the ratio \( \frac{PS}{PQ} = \frac{6.6}{30} = 0.22 \) and \( \frac{PT}{PR} = \frac{5.5}{25} = 0.22 \). Since the ratios are equal, by Thales' theorem, \( \overline{ST} \parallel \overline{QR} \) in the first triangle.

Step3: Analyze the second triangle

- \( PS = 7.5 \), \( PQ = PS + SQ = 7.5 + 17.5 = 25 \)
- \( PT = 8.7 \), \( PR = PT + TR = 8.7 + 20.3 = 29 \)
- Check the ratio \( \frac{PS}{PQ} = \frac{7.5}{25} = 0.3 \) and \( \frac{PT}{PR} = \frac{8.7}{29} = 0.3 \). Wait, let's recalculate: \( 7.5\div25 = 0.3 \), \( 8.7\div29 = 0.3 \)? Wait, \( 29\times0.3 = 8.7 \), yes. Wait, but wait, maybe I made a mistake. Wait, \( PS = 7.5 \), \( SQ = 17.5 \), so \( PQ = 7.5 + 17.5 = 25 \). \( PT = 8.7 \), \( TR = 20.3 \), so \( PR = 8.7 + 20.3 = 29 \). Now, \( \frac{PS}{SQ} = \frac{7.5}{17.5} = \frac{3}{7} \approx 0.428 \), \( \frac{PT}{TR} = \frac{8.7}{20.3} \approx 0.428 \). Wait, maybe I used the wrong ratio. Let's check \( \frac{PS}{PQ} = \frac{7.5}{25} = 0.3 \), \( \frac{PT}{PR} = \frac{8.7}{29} = 0.3 \). Wait, but let's check the third triangle.

Step4: Analyze the third triangle

- \( PS = 9 \), \( PQ = PS + SQ = 9 + 16 = 25 \)
- \( PT = 11.2 \), \( PR = PT + TR = 11.2 + 17.8 = 29 \)
- Check the ratio \( \frac{PS}{PQ} = \frac{9}{25} = 0.36 \), \( \frac{PT}{PR} = \frac{11.2}{29} \approx 0.386 \). These ratios are not equal, so \( \overline{ST} \) is not parallel to \( \overline{QR} \) here.

Wait, but in the second triangle, let's recalculate \( \frac{PS}{PQ} = \frac{7.5}{25} = 0.3 \), \( \frac{PT}{PR} = \frac{8.7}{29} = 0.3 \). Wait, but \( 7.5 + 17.5 = 25 \), \( 8.7 + 20.3 = 29 \). Wait, but \( 7.5/17.5 = 3/7 \approx 0.428 \), \( 8.7/20.3 = 87/203 = 3/7 \) (since \( 87\div29 = 3 \), \( 203\div29 = 7 \)). Oh! So \( \frac{PS}{SQ} = \frac{PT}{TR} = \frac{3}{7} \), which also satisfies the theorem (since if a line divides two sides of a triangle proportionally, then it is parallel to the third side). So both the first and second triangles? Wait, no, wait the first triangle: \( PS = 6.6 \), \( SQ = 23.4 \), so \( \frac{PS}{SQ} = 6.6/23.4 = 66/234 = 11/39 \approx 0.282 \). \( PT = 5.5 \), \( TR = 19.5 \), so \( \frac{PT}{TR} = 5.5/19.5 = 55/195 = 11/39 \approx 0.282 \). So \( \frac{PS}{SQ} = \frac{PT}{TR} \), so \( ST \parallel QR \). The second triangle: \( \frac{PS}{SQ} = 7.5/17.5 = 3/7 \), \( \frac{PT}{TR} = 8.7/20.3 = 87/203 = 3/7 \) (since \( 87 = 29\times3 \), \( 203 = 29\times7 \)), so \( \frac{PS}{SQ} = \frac{PT}{TR} \), so \( ST \parallel QR \). Wait, but the third triangle: \( PS = 9 \), \( SQ = 16 \), \( PT = 11.2 \), \( TR = 17.8 \). \( 9/16 = 0.5625 \), \( 11.2/17.8 = 56/89 \approx 0.629 \), not equal. So the first and second triangles? Wait, but the original problem's first triangle: \( PS = 6.6 \), \( SQ = 23.4 \), \( PT = 5.5 \), \( TR = 19.5 \). \( 6.6/23.4 = 66/234 = 11/39 \), \( 5.5/19.5 = 55/195 = 11/39 \). So that's equal. The second triangle: \( 7.5/17.5 = 3/7 \), \( 8.7/20.3 = 3/7 \). So both first and second? Wait, but maybe I made a mistake in the third. Wait, the third triangle: \( PS = 9 \), \( SQ = 16 \), so \( PQ = 25 \); \( PT = 11.2 \), \( TR = 17.8 \), so \( PR = 29 \). \( 9/25 = 0.36 \), \( 11.2/29 \approx 0.386 \), not equal. So the first and second figures show \( \overline{ST} \parallel \overline{QR} \). Wait, but let's check the first triangle again: \( PS = 6.6 \), \( PQ = 6.6 + 23.4 = 30 \); \( PT = 5.5 \), \( PR = 5.5 + 19.5 = 25 \). \( 6.6/30 = 0.22 \), \( 5.5/25 = 0.22 \). So that's equal. The second triangle: \( 7.5/25 = 0.3 \), \( 8.7/29 = 0.3 \). So that's equal. So both first and second. Wait, but maybe the problem is that in the second triangle, it's a trapezoid? No, it's a triangle. Wait, the first figure is a right triangle, the second is an isoceles? No, the second has sides 7.5, 17.5 and 8.7, 20.3. Wait, but according to Thales' theorem, if the line divides the two sides proportionally, then it's parallel. So in the first figure: \( \frac{PS}{PQ} = \frac{6.6}{30} = 0.22 \), \( \frac{PT}{PR} = \frac{5.5}{25} = 0.22 \). So equal. In the second figure: \( \frac{PS}{PQ} = \frac{7.5}{25} = 0.3 \), \( \frac{PT}{PR} = \frac{8.7}{29} = 0.3 \). So equal. In the third figure: \( \frac{PS}{PQ} = \frac{9}{25} = 0.36 \), \( \frac{PT}{PR} = \frac{11.2}{29} \approx 0.386 \). Not equal. So the first and second figures.

Answer:

The first and second figures (the top two triangles) show that \( \overline{ST} \parallel \overline{QR} \).