Question

in the diagram below, \\( \overline { s t } \\) is parallel to \\( \overline { p q } \\). if \\( s t \\) is 6 more than \\( r t \\), \\( r q = 9 \\), and \\( p q = 18 \\), find the length of \\( \overline { r t } \\). figures are not necessarily drawn to scale. state your answer in simplest radical form, if necessary.

Explanation:

Step1: Use the similarity of triangles

Since \( \overline{ST}\parallel\overline{PQ}\), then \(\triangle RST\sim\triangle RPQ\) (by the AA similarity criterion, as \(\angle R=\angle R\) (common angle) and \(\angle RST=\angle RPQ\) (corresponding angles for parallel lines \(ST\) and \(PQ\))).
For similar triangles \(\triangle RST\) and \(\triangle RPQ\), the ratio of their corresponding sides is equal. That is \(\frac{ST}{PQ}=\frac{RT}{RQ}\).
Let \(RT = x\). Then \(ST=x + 6\), \(RQ = 9\), and \(PQ=18\).
Substituting these values into the proportion \(\frac{ST}{PQ}=\frac{RT}{RQ}\), we get \(\frac{x + 6}{18}=\frac{x}{9}\).

Step2: Cross - multiply and solve the equation

Cross - multiply the equation \(\frac{x + 6}{18}=\frac{x}{9}\):

$$9(x + 6)=18x$$

Expand the left - hand side: \(9x+54 = 18x\).
Subtract \(9x\) from both sides: \(54=18x-9x\).
Simplify the right - hand side: \(9x = 54\).
Divide both sides by 9: \(x=\frac{54}{9}=6\).

Answer:

The length of \(\overline{RT}\) is \(6\).