To determine the gradient of a line perpendicular to a given line, we need to understand the relationship between their slopes. If two lines are perpendicular, the product of their gradients (slopes) is [tex]\(-1\)[/tex].
Given that the gradient (slope) of line [tex]\(A\)[/tex] is [tex]\(4\)[/tex], we denote this as:
[tex]\[ m_A = 4 \][/tex]
Let the gradient of the line perpendicular to [tex]\(A\)[/tex] be [tex]\(m_P\)[/tex]. According to the property of perpendicular lines, we have:
[tex]\[ m_A \times m_P = -1 \][/tex]
Substituting the known gradient of line [tex]\(A\)[/tex] into this equation:
[tex]\[ 4 \times m_P = -1 \][/tex]
To find [tex]\(m_P\)[/tex], we solve for [tex]\(m_P\)[/tex] by dividing both sides of the equation by [tex]\(4\)[/tex]:
[tex]\[ m_P = \frac{-1}{4} \][/tex]
Thus, the gradient of the line which is perpendicular to [tex]\(A\)[/tex] is:
[tex]\[ m_P = -0.25 \][/tex]
