Question

p
9
11.2
s
t
16
17.8
q
r
p
7.5
8.7
s
t
17.5
20.3
q
r

Explanation:

Step1: Check the ratio of the segments for the first triangle

For the first triangle, calculate the ratios \(\frac{PS}{SQ}=\frac{9}{16}\) and \(\frac{PT}{TR}=\frac{11.2}{17.8}\).
\(\frac{11.2}{17.8}=\frac{112}{178}=\frac{56}{89}\approx0.63\), \(\frac{9}{16} = 0.5625\). Since \(\frac{PS}{SQ}
eq\frac{PT}{TR}\), \(ST\) is not parallel to \(QR\) in the first triangle.

Step2: Check the ratio of the segments for the second triangle

For the second triangle, calculate the ratios \(\frac{PS}{SQ}=\frac{7.5}{17.5}=\frac{75}{175}=\frac{3}{7}\approx0.43\) and \(\frac{PT}{TR}=\frac{8.7}{20.3}=\frac{87}{203}=\frac{3}{7}\) (because \(87 = 3\times29\) and \(203=7\times29\)). Since \(\frac{PS}{SQ}=\frac{PT}{TR}\), by the converse of the basic proportionality theorem (Thales' theorem), \(ST\parallel QR\) in the second triangle.

Answer:

The second triangle (the one with \(PS = 7.5\), \(SQ=17.5\), \(PT = 8.7\), \(TR = 20.3\)) has \(ST\parallel QR\).