Question

given: \\( \overleftrightarrow{q s} \perp \overleftrightarrow{u s} \\) and \\( \triangle q r s \sim \triangle s t u \\) prove: slope of \\( \overleftrightarrow{q s} \times \\) slope of \\( \overleftrightarrow{u s}=-1 \\) the table shows the proof of the relationship between the slopes of two perpendicular lines. what is the missing statement in step 5? a. slope of \\( \overleftrightarrow{q s} \times \\) slope of \\( \overleftrightarrow{u s}=\frac{s t}{t u} \times-\frac{t u}{s t} \\) b. slope of \\( \overleftrightarrow{q s} \times \\) slope of \\( \overleftrightarrow{u s}=\frac{s t}{t u} \times \frac{t u}{s t} \\) c. slope of \\( \overleftrightarrow{q s} \times \\) slope of \\( \overleftrightarrow{u s}=1 \\) d. slope of \\( \overleftrightarrow{q s} \times(- \\) slope of \\( \overleftrightarrow{u s})=1 \\)

Explanation:

Step1: Analyze step 2 and step 4

From step 2, we have \(\frac{QR}{RS}=\frac{ST}{TU}\). In step 4, we have \(slope\ of\ \overleftrightarrow{QS}\times slope\ of\ \overleftrightarrow{US}=\frac{QR}{RS}\times(-\frac{TU}{ST})\).

Step2: Substitute using step 2

Since \(\frac{QR}{RS}=\frac{ST}{TU}\), substitute \(\frac{QR}{RS}\) in the product of slopes formula. So \(slope\ of\ \overleftrightarrow{QS}\times slope\ of\ \overleftrightarrow{US}=\frac{ST}{TU}\times(-\frac{TU}{ST})\)

Answer:

A. \(slope\ of\ \overleftrightarrow{QS}\times slope\ of\ \overleftrightarrow{US}=\frac{ST}{TU}\times(-\frac{TU}{ST})\)