This section provides an exploration of perpendicular lines in coordinate geometry, emphasizing calculations and equations over textual explanations. We'll cover how to determine the gradient and equation of lines perpendicular to a given line, including the method for finding the perpendicular bisector of a line segment.

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Understanding Gradients and Perpendicularity

Gradient formula:

Perpendicular condition:

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Example 1: Gradient of a Perpendicular Line

Given a line with equation $y = 2x + 5$, determine the gradient of a line perpendicular to it.

• Original gradient: $m = 2$
• Perpendicular gradient: $m_{\perp} = -\frac{1}{m} = -0.5$

Thus, the perpendicular gradient is $-0.5$.

Equation of Perpendicular Lines

Example 2: Finding Equation of Perpendicular Line

Determine the equation of a line perpendicular to $y = -3x + 1$ that passes through $(4, 3)$.

• Given gradient: $m = -3$
• Perpendicular gradient: $m_{\perp} = \frac{1}{3}$
• Point: $(4, 3)$
• Equation: $y - y_1 = m{\perp}(x - x_1)$
• Substituting values:

$y - 3 = \frac{1}{3} (x - 4)$

$y = \frac{1}{3} - \frac{4}{3} + 3$

$y = \frac{1}{3}x + \frac{5}{3}$

Thus, the equation is $y = \frac{1}{3}x + \frac{5}{3}$.

Graphical Representation:

Finding the Perpendicular Bisector

Example 3: Perpendicular Bisector Between Points $A(2, -1)$ and $B(6, 3)$

• Midpoint of $AB$: $M = (4, 1)$
• Gradient of $AB$: $m_{AB} = 1$
• Perpendicular gradient: $m_{\perp} = -1$
• Using $M$ and $m_{\perp}$, equation: $y = -x + 5$

The perpendicular bisector of the segment is $y = -x + 5$.

Graphical Representation: