Question

1. choose the correct answer.

given \\( \overleftrightarrow { q r }, \overleftrightarrow { p t }, \overleftrightarrow { o s }, \overleftrightarrow { q t } \\), which pair of lines is perpendicular?
\\( \overleftrightarrow { q t } \\) and \\( \overleftrightarrow { o s } \\)
\\( \overleftrightarrow { q r } \\) and \\( \overleftrightarrow { o s } \\)
\\( \overleftrightarrow { q t } \\) and \\( \overleftrightarrow { p t } \\)
\\( \overleftrightarrow { p t } \\) and \\( \overleftrightarrow { o s } \\)

Explanation:

Step1: Recall the property of perpendicular lines

Perpendicular lines intersect at a right - angle (\(90^{\circ}\)). In a cube (the given figure is a cube - like structure), we use the geometric property of lines in 3 - D space.

Step2: Analyze each option

• Option 1: \(\overleftrightarrow{QT}\) and \(\overleftrightarrow{OS}\)

Since \(QT\) and \(OS\) are parallel (in a cube, opposite edges of the base - like face are parallel), they are not perpendicular.

• Option 2: \(\overleftrightarrow{QR}\) and \(\overleftrightarrow{OS}\)

\(QR\) and \(OS\) are skew lines (not in the same plane and do not intersect). They are not perpendicular.

• Option 3: \(\overleftrightarrow{QT}\) and \(\overleftrightarrow{PT}\)

\(QT\) is a vertical - like edge and \(PT\) is a horizontal - like edge. They intersect at a right - angle.

• Option 4: \(\overleftrightarrow{PT}\) and \(\overleftrightarrow{OS}\)

\(PT\) and \(OS\) are parallel (in a cube, opposite edges of the top - like face are parallel), they are not perpendicular.

Answer:

\(\overleftrightarrow{QT}\) and \(\overleftrightarrow{PT}\) (the third option)