Question

given that line s is perpendicular to line t, which statements must be true of the two lines? check all that apply.

lines s and t have slopes that are opposite reciprocals.

lines s and t have the same slope.

the product of the slopes of s and t is equal to −1.

the lines have the same steepness.

the lines have different y intercepts.

the lines never intersect.

the intersection of s and t forms right angle.

if the slope of s is 6, the slope of t is −6.

Explanation:

1. For two non - vertical perpendicular lines, the slopes are opposite reciprocals. If the slope of line \(s\) is \(m_1\) and the slope of line \(t\) is \(m_2\), then \(m_1=-\frac{1}{m_2}\) (or \(m_1\times m_2=- 1\)) when both lines are non - vertical.
2. Lines with the same slope are parallel, not perpendicular, so the statement "Lines \(s\) and \(t\) have the same slope" is false.
3. From the definition of perpendicular lines (for non - vertical lines), if \(m_1\) and \(m_2\) are the slopes of two perpendicular lines, then \(m_1\times m_2=-1\).
4. The steepness of a line is related to the absolute value of the slope. For perpendicular lines with slopes \(m_1\) and \(m_2\) (where \(m_1\times m_2 = - 1\)), if \(|m_1|

eq1\), then \(|m_1|
eq|m_2|\). For example, if \(m_1 = 2\), then \(m_2=-\frac{1}{2}\), and \(|2|
eq|-\frac{1}{2}|\), so they do not have the same steepness.

1. The \(y\) - intercepts of two perpendicular lines can be the same or different. For example, the lines \(y = x\) and \(y=-x\) are perpendicular and have the same \(y\) - intercept (\(0\)), and the lines \(y=x + 1\) and \(y=-x+2\) are perpendicular and have different \(y\) - intercepts. So we cannot say that the lines must have different \(y\) - intercepts.
2. Perpendicular lines (in a plane) intersect at a right angle, so they do intersect. The statement "The lines never intersect" is false (this is a property of parallel lines).
3. By the definition of perpendicular lines, when two lines are perpendicular, the angle between them is \(90^{\circ}\) (a right angle).
4. If the slope of \(s\) is \(6\), the slope of \(t\) should be \(-\frac{1}{6}\) (since \(6\times(-\frac{1}{6})=-1\)) not \(-6\).

Answer:

• Lines \(s\) and \(t\) have slopes that are opposite reciprocals.
• The product of the slopes of \(s\) and \(t\) is equal to \(-1\).
• The intersection of \(s\) and \(t\) forms a right angle.