Question

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

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' Theorem)

The theorem states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side. For the first figure:
$$\frac{PS}{SQ}=\frac{8.4}{21.6}=\frac{84}{216}=\frac{7}{18}$$
$$\frac{PT}{TR}=\frac{7}{18}$$
Since $\frac{PS}{SQ}=\frac{PT}{TR}$, by the Basic Proportionality Theorem, $\overline{ST}\parallel\overline{QR}$.

Step2: Check the second figure using the same theorem

For the second figure:
$$\frac{PS}{SQ}=\frac{6.6}{23.4}=\frac{66}{234}=\frac{11}{39}$$
$$\frac{PT}{TR}=\frac{5.5}{19.5}=\frac{55}{195}=\frac{11}{39}$$
Since $\frac{PS}{SQ}=\frac{PT}{TR}$, by the Basic Proportionality Theorem, $\overline{ST}\parallel\overline{QR}$.

Answer:

Both figures show that $\overline{ST}\parallel\overline{QR}$.