Question

which figures show that st || qr?

Explanation:

Step1: Apply the converse of the basic proportionality theorem (Thales' theorem)

The converse of Thales' theorem states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. For \(\triangle PQR\) with line \(ST\), we check the ratios \(\frac{PS}{SQ}\) and \(\frac{PT}{TR}\).

For the first triangle:

• Calculate \(\frac{PS}{SQ}=\frac{6.6}{23.4 - 6.6}=\frac{6.6}{16.8}=\frac{66}{168}=\frac{11}{28}\)
• Calculate \(\frac{PT}{TR}=\frac{5.5}{19.5 - 5.5}=\frac{5.5}{14}=\frac{55}{140}=\frac{11}{28}\)

Since \(\frac{PS}{SQ}=\frac{PT}{TR}\), by the converse of Thales' theorem, \(ST\parallel QR\) in the first triangle.

For the second triangle:

• Calculate \(\frac{PS}{SQ}=\frac{9}{16}\)
• Calculate \(\frac{PT}{TR}=\frac{11.2}{17.8}=\frac{112}{178}=\frac{56}{89}\)

Since \(\frac{PS}{SQ}
eq\frac{PT}{TR}\), \(ST\) is not parallel to \(QR\) in the second triangle.

Answer:

The first figure shows that \(ST\parallel QR\)