Question

given the work below, what can you conclude about the relationship between \\( \overline { s t } \\) and \\( \overline { p r } \\)?
\\( \frac { q s } { s p } = \frac { q t } { t r } \\)
\\( \frac { 7 } { 11.2 } = \frac { 10 } { 16 } \\)
\\( 7 ( 16 ) = 10 ( 11.2 ) \\)
\\( 112 = 112 \\)
\\( \overline { s t } \\) is parallel to \\( \overline { p r } \\)
\\( \overline { s t } \\) is congruent to \\( \overline { p r } \\)
there is no relationship between \\( \overline { s t } \\) and \\( \overline { p r } \\)
\\( \overline { s t } \\) is half the length of \\( \overline { p r } \\)

Explanation:

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

If a line divides two sides of a triangle proportionally, then the line is parallel to the third side.

Step2: Check the proportion

We are given \(\frac{QS}{SP}=\frac{7}{11.2}\) and \(\frac{QT}{TR}=\frac{10}{16}\).
Simplify \(\frac{7}{11.2}=\frac{70}{112}=\frac{5}{8}\) and \(\frac{10}{16}=\frac{5}{8}\).
Since \(\frac{QS}{SP}=\frac{QT}{TR}\), by the converse of the basic proportionality theorem.

Answer:

\(\overline{ST}\) is parallel to \(\overline{PR}\)